Modern Theory of Magnetism in Metals and Alloys

Springer Series in

solid-state sciences

175

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Yoshiro Kakehashi

Modern Theory of Magnetism in Metals and Alloys

Yoshiro Kakehashi Faculty of Science, Department of Physics University of the Ryukyus Nishihara, Japan Series Editors: Professor Dr., Dres. h. c. Manuel Cardona Professor Dr., Dres. h. c. Peter Fulde* Professor Dr., Dres. h. c. Klaus von Klitzing Professor Dr., Dres. h. c. Hans-Joachim Queisser Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany *Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38 01187 Dresden, Germany Professor Dr. Roberto Merlin Department of Physics, University of Michigan 450 Church Street, Ann Arbor, MI 48109-1040, USA Professor Dr. Horst Störmer Dept. Phys. and Dept. Appl. Physics, Columbia University, New York, NY 10027, USA and Bell Labs., Lucent Technologies, Murray Hill, NJ 07974, USA

ISSN 0171-1873 Springer Series in Solid-State Sciences ISBN 978-3-642-33400-9 ISBN 978-3-642-33401-6 (eBook) DOI 10.1007/978-3-642-33401-6 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012954262 © Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Metallic magnetism has a long history because there have been continuous discoveries of many intriguing phenomena and difficulties in their theoretical description. One of the long-standing problems has been known as the itinerant vs localized behavior of the magnetism. The ground-state properties of Fe, Co, and Ni such as magnetization and the T -linear specific heat at low temperatures, for example, are explained by the band model, while their finite temperature properties such as the paramagnetic susceptibility and the large specific heat anomaly at the Curie temperature are explained well by the localized model. The dual property of metallic magnetism led to two paths in theoretical investigations. One is to develop the band theory at the ground state taking into account correlation effects on the one electron potential for electrons. There the density functional theory (DFT) has played an important role. Theoretical improvement of metallic magnetism at the ground state has been achieved as a part of the developments of the DFT in the electronic structure calculations. Another direction of the development has been to take into account the spin fluctuations in order to describe local-moment behaviors of metallic magnetism at finite temperatures. Theoretical results in this direction until 1980 are summarized in the book by Moriya (Spin Fluctuations in Itinerant Electron Magnetism (Springer, Berlin, 1985)). Although spin fluctuation theories have succeeded in describing the local moment behavior at finite temperatures in metallic magnetism, the underlying electronic structure related to the magnetism of a certain individual material seems to be oversimplified. A book which unifies the two paths on the same footing would be valuable for readers to understand the metallic magnetism. This book aims to describe the theories of metallic magnetism from both viewpoints, namely spin fluctuations and the electronic structure. It attempts to clarify the magnetism from metals to disordered alloys to amorphous alloys. The book covers most of the traditional topics of metallic magnetism such as electron correlation effects on the ferromagnetism, magnetic excitations, as well as the stability of antiferromagnetism and spin density waves. But it also includes topics which have been developed in the past three decades. The first is the development of the dynamical CPA (coherent potential approximation), which describes v

vi

Preface

the dynamical spin and charge fluctuations on the basis of the microscopic electronic structure within the single-site approximation. In particular, the first-principles dynamical CPA has reduced the gap between the spin fluctuation theory and the band theoretical approach to a large extent, thus allowing the investigation of the relationship between metallic magnetism and electronic structure. We elucidate this theory in Chap. 3. We also point out in the same chapter that the dynamical CPA is equivalent to the dynamical mean-field theory (DMFT) in the metal-insulator transition. The second topic is the theory of local environment effects (LEE) in disordered alloys, which goes beyond the single-site CPA theory of magnetism. In Chap. 8, we describe the theory and clarify the magnetic behavior in the vicinity of the magnetic instability of Fe–Ni, Ni–Mn, and Ni–Cu alloys. This chapter also includes the molecular dynamics approach, which automatically determines the complex magnetic structure in metals and alloys. The third topic is the theoretical development of magnetism in amorphous metals and alloys. The finite-temperature theory sheds light on the amorphous magnetism from the viewpoint of spin fluctuations and the LEE, and clarifies how structural disorder drastically changes the magnetic properties of metals and alloys. This development is discussed in Chap. 9. Chapter 1 presents an introduction for the readers who are not familiar with the magnetism. The frustrated system with heavy effective mass (e.g., YMn2 and LiV2 O4 ) is not described in this book, because it is still under development. Recent topics on the spintronics are also omitted for the same reason. Non-local theory of dynamical spin fluctuations which goes beyond the dynamical CPA is left as a problem of future concern. I would like to express my sincere thanks to Professor Peter Fulde, Professor Matin C. Gutzwiller, and Professor Hiroshi Miwa for their continuous support and encouragement over 30 years. I would like to thank Professor Takashi Uchida and Professor Ming Yu for their critical reading of the manuscript as well as their valuable comments; Professor Shi-Yu Wu and Professor Sung G. Chung for their valuable suggestions for improvement; and Professor Takeo Fujiwara and Professor Mojmir Sob for their kind comments. Dr. M.A.R. Patoary kindly prepared useful figures to whom I am most indebted. Thanks are also given to Mr. K. Chung for his help in proofreading the manuscript. Finally, I am grateful to Dr. C. Ascheron, Publishing Editor of Springer Verlag for his support towards the publication of this book. Okinawa July 2012

Y. Kakehashi

Contents

1

Introduction to Magnetism . . . . . . . . . 1.1 Magnetic Moments . . . . . . . . . . . 1.2 Basic Hamiltonian . . . . . . . . . . . . 1.3 Formation of Atomic Moments . . . . . 1.4 Metal and Insulator in Solids . . . . . . 1.5 Quenching of Orbital Magnetic Moments 1.6 Heisenberg Model . . . . . . . . . . . . 1.7 Magnetic Structure . . . . . . . . . . . .

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1 1 3 6 11 18 19 22

2

Metallic Magnetism at the Ground State . . . . . . . . . . . . . 2.1 Band Theory of Ferromagnetism . . . . . . . . . . . . . . . 2.2 Electron Correlations on Magnetism . . . . . . . . . . . . . 2.2.1 Stoner Condition in the Correlated Electron System . 2.2.2 Stability of Ferromagnetism in the Low Density Limit 2.2.3 Gutzwiller Theory of Electron Correlations . . . . . . 2.3 Density Functional Approach . . . . . . . . . . . . . . . . . 2.3.1 Slater’s Band Theory . . . . . . . . . . . . . . . . . 2.3.2 Density Functional Theory . . . . . . . . . . . . . . 2.3.3 Tight-Binding Linear Muffin-Tin Orbitals . . . . . . 2.3.4 Ferromagnetism in Transition Metals . . . . . . . . .

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29 29 36 36 37 40 47 47 48 53 58

3

Metallic Magnetism at Finite Temperatures . . . . . . . . . . . . 3.1 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Functional Integral Method . . . . . . . . . . . . . . . . . . . 3.3 Single-Site Theory in the Static Approximation . . . . . . . . . 3.4 Dynamical CPA Theory . . . . . . . . . . . . . . . . . . . . . 3.5 Dynamical CPA with Harmonic Approximation . . . . . . . . 3.6 Dynamical CPA and Dynamical Mean-Field Theory . . . . . . 3.6.1 The Many-Body CPA and Its Equivalence to the Dynamical CPA . . . . . . . . . . . . . . . . . . 3.6.2 The DMFT and Its Equivalence to the Dynamical CPA .

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63 63 67 74 83 90 95

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95 96 vii

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Contents

3.6.3

The Projection Operator Method CPA and Summary of Relations . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.7 First-Principles Dynamical CPA and Metallic Magnetism . . . . . 104 4

Magnetic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Spin Waves in the Local Moment System . . . . . . . . . . . . . 4.2 Spin Waves in Itinerant Ferromagnets . . . . . . . . . . . . . . . 4.3 Dynamical Susceptibility . . . . . . . . . . . . . . . . . . . . . 4.4 Dynamical Susceptibility in the RPA and Spin Wave Excitations .

5

Spin Fluctuation Theory in Weak Ferromagnets . . . . . . . . . . . . 135 5.1 Free-Energy Formulation of the Stoner Theory . . . . . . . . . . . 135 5.2 Self-Consistent Renormalization Theory . . . . . . . . . . . . . . 140

6

Antiferromagnetism and Spin Density Waves . . . . . . . . . . . 6.1 Antiferromagnetism in Metals . . . . . . . . . . . . . . . . . . 6.2 Generalized Static Susceptibility and Antiferromagnetism . . . 6.3 Molecular Dynamics Theory for Complex Magnetic Structures 6.4 Phenomenological Theory of Magnetic Structure . . . . . . . . 6.4.1 Multiple SDW with Commensurate Wave Vectors . . . 6.4.2 Multiple SDW with Incommensurate Wave Vectors . .

7

Magnetism in Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . 181 7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys . . . . . . 181 7.2 Magnetic Impurity in Noble Metals . . . . . . . . . . . . . . . . . 193

8

Magnetism of Disordered Alloys . . . . . . . . . . . . . . . 8.1 Slater–Pauling Curves . . . . . . . . . . . . . . . . . . . 8.2 Single-Site Theory of Disordered Alloys . . . . . . . . . 8.3 Theory of Local Environment Effects in Magnetic Alloys 8.4 Computer Simulations for Disordered Magnetic Alloys .

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203 203 206 217 242

9

Magnetism of Amorphous Metals and Alloys . . . . . . . . . . . . 9.1 Introduction to the Amorphous Metallic Magnetism . . . . . . . 9.2 Amorphous Structure and Electronic Structure . . . . . . . . . . 9.3 Theory of Amorphous Metallic Magnetism . . . . . . . . . . . . 9.4 Magnetism of Amorphous Transition Metals . . . . . . . . . . . 9.4.1 General Survey . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Magnetism of Amorphous Fe, Co, and Ni . . . . . . . . . 9.4.3 Degree of Structural Disorder and Nonunique Magnetism 9.5 Theory of Magnetism in Amorphous Alloys . . . . . . . . . . . 9.6 Magnetism of Amorphous Transition Metal Alloys . . . . . . . . 9.6.1 TM–TM Amorphous Alloys . . . . . . . . . . . . . . . . 9.6.2 The Other TM Alloys . . . . . . . . . . . . . . . . . . .

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253 253 255 258 266 267 270 276 280 287 287 295

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115 115 118 124 129

149 149 157 162 169 172 177

Appendix A Equivalence of the CPA Equations (3.83), (3.85), and (3.89) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Appendix B

Dynamical CPA Based on the Multiple Scattering Theory . 305

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ix

Appendix C Derivation of the Single-Site Spin Fluctuation Theory from the Dynamical CPA . . . . . . . . . . . . . . . . . . . . . . . . 309 Appendix D Expansion of Dνσ with Respect to Dynamical Potential . . 313 Appendix E

Linear Response Theory . . . . . . . . . . . . . . . . . . . 319

Appendix F Isothermal Molecular Dynamics and Canonical Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Appendix G Recursion Method for Electronic Structure Calculations . 325 Appendix H An Integral in the RKKY Interaction . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Chapter 1

Introduction to Magnetism

Magnetic properties originate in the spin degrees of freedom of electrons and their associated motion in solids. We first describe the microscopic magnetic moments of electrons, and the formation of atomic magnetic moments due to strong Coulomb interactions in an atom with unfilled shell. Atomic magnetic moments change their nature when the atoms form a solid. The key to understanding the behavior of magnetic moments in solids is the degree of electron localization. We briefly introduce the concept of the metal and the insulator (i.e., the Mott insulator). Electrons in the latter are localized on each atom, so that their magnetic properties are described by the atomic magnetic moments and the magnetic interactions between them. We present in this chapter only a minimal discussion on the magnetism in insulators. On the other hand, electrons move from site to site over the entire crystal in metals, so that the magnetic properties are connected to the whole degrees of freedom of correlated electrons. Needless to say, the main theme of this book is the magnetism of metals and alloys. In the last section, we elucidate various magnetic structures in solids to provide a basic knowledge on magnetism.

1.1 Magnetic Moments Magnetic moment M in the electromagnetics is defined by the torque N on a magnet under the magnetic field H as N ≡ M × H.

(1.1)

It originates in electrons in the magnet. According to the classical electromagnetics, the coil-type local current i(r) due to electron motion causes a magnetic moment. It is given in the CGS-Gauss unit as [1] ML =

1 2c

 r × i(r) d 3 x.

Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_1, © Springer-Verlag Berlin Heidelberg 2012

(1.2) 1

2

1

Introduction to Magnetism

Here c is the speed of light. i(r) is the current density of electrons, given by i(r) =  − i ev i δ(r − r i ) with v i being the velocity of the electron and −e the charge of the electron at r i . Substituting the expressions into (1.2), we obtain ML = −

 i

e li . 2me c

(1.3)

Here me is the mass of an electron and l i denotes the angular momentum of electron i. Equation (1.3) indicates that an electron with angular momentum l generates a magnetic moment, ml = −

e l. 2me c

(1.4)

This is called the orbital magnetic moment for an electron. According to the quantum mechanics, an electron has its own magnetic moment called the spin magnetic moment [2]. It is given by ms = −ge

e s. 2me c

(1.5)

Here s is the spin angular momentum of an electron with spin s = 1/2. The constant ge = 2.0023 is referred as the g-value of electron. The deviation from 2 is caused by the interaction with electro-magnetic fields. The spin magnetic moment was found first experimentally by Stern and Gerlach in 1922, and was established theoretically in 1928 by Dirac in his relativistic theory of electrons [3]. In the following we assume that ge = 2 for simplicity. As seen from (1.5), the spin of an electron leads to the magnetic moment μB =

e = 0.9274 × 10−20 emu. 2me c

(1.6)

Here  is the Planck constant divided by 2π . The magnetic moment μB denotes the Bohr magneton, and is often used as a unit of the magnetic moment in atomic scale. Equations (1.4) and (1.5) indicate that an electron has the following magnetic moment in the atomic scale. m = ml + ms = −(l + 2s)μB ,

(1.7)

and the total magnetic moment is given by   M =− (l i + 2s i )μB . i

(1.8)

1.2 Basic Hamiltonian

3

Here the angular momenta l and s are measured in units of  = 1. The l i and s i at the right-hand-side (r.h.s.) of the above equation should be regarded as quantum mechanical operators, and ∼ means a quantum mechanical expectation value. In the following, we omit the minus sign at the r.h.s. of (1.8) for convenience bearing in mind that the real magnetic moments are opposite in direction. It should be noted that the nuclei in the magnet also have magnetic moments because they have their own spins. The size of their magnetic moments is however characterized by the nuclear magneton μN , which is defined by μN =

e . 2mp c

(1.9)

Here mp denotes the mass of proton. The nuclear magneton μN is only 1/1800 of the Bohr magneton μB because μN /μB = me /mp . Therefore the magnetism in solids is dominated by the magnetic moments of electrons. In the following equations, we adopt a unit of μB = 1 for simplicity.

1.2 Basic Hamiltonian Magnetic moments and magnetic properties are governed by the Hamiltonian of the system. The basic Hamiltonian of electrons in solids is given as follows in the second quantization.   1 ψ † (r) − ∇ 2 + vN (r) ψ(r) dr 2 

1 1 ψ r  ψ(r) dr dr  + ψ † (r)ψ † r   2 |r − r |   1 1 + ψ † (r) ∇vN (r) × (−i∇) · σ ψ(r) dr 2 2  − ψ † (r)(l + 2s)ψ(r) dr · H .

 H=

(1.10)

Here we have adopted the units me = 1,  = 1, and e = 1 for simplicity. ψ(r) = (ψ↑ (r), ψ↓ (r)) is the electron field operator.  vN (r) is the one electron potential energy due to nuclei, and is given by vN (r) = − i Zi /|r − R i |. Zi is the atomic number of the atom at position R i . H denotes the magnetic field, and σ = (σx , σy , σz ) denote the Pauli spin matrices given by 

 0 1 σx = , 1 0



 0 −i σy = , i 0



 1 0 σz = . 0 −1

(1.11)

The first term in the Hamiltonian (1.10) consists of the kinetic energy and the attractive potential due to nuclei, and describes the independent motion of electrons.

4

1

Introduction to Magnetism

The second term denotes the electron–electron interaction, the third term denotes the spin–orbit interaction, and the last term is the Zeeman interaction due to magnetic field. Note that in the above expression we have omitted the diamagnetic term which is proportional to the square of the magnetic field [2], because we do not consider the diamagnetism in this book. In solids, we may express the field operators by means of a basis set of wavefunctions {ϕi (r)} as follows.  ψσ (r) = aiσ ϕi (r). (1.12) i

Here ϕi (r) are orthonormal basis functions. The suffix i stands for a set of the atomic position i (quantum number n) and orbital L (momentum k) when we adopt atomic orbitals (one-electron eigen functions in solids) as the basis functions. Using the orthonormal basis set, we can express the Hamiltonian (1.10) as follows.  1  † † † εij aiσ aj σ + Vij kl aiσ aj σ  akσ  alσ + HSO + HZeeman . (1.13) H= 2  ij σ

ij klσ σ

Here εij (Vij kl ) are the matrix elements for the independent electron (the Coulomb interaction) part of the Hamiltonian (1.10). They are given by    1 2 ∗ (1.14) εij = dr ϕi (r) − ∇ + vN (r) ϕj (r), 2  ϕi∗ (r)ϕj∗ (r  )ϕk (r  )ϕl (r) . (1.15) Vij kl = dr dr  |r − r  | The third and last terms at the r.h.s. of (1.13) are the spin–orbit interaction term and the Zeeman term, respectively.  † ζiαj γ aiα aj γ , (1.16) HSO = iαj γ

HZeeman = −



† (l + 2s)iαj γ aiα aj γ · H .

(1.17)

iαj γ

Here the spin–orbit interaction matrix elements ζiαj γ are given by   1 ∗ ζiαj γ = dr ϕi (r) ∇v(r) × (−i∇) ϕj (r) · (s)αγ . 2

(1.18)

In molecules and solids, atomic orbitals are often useful as the basis functions.

(atom) ϕi (r) = φinl |r − R i | Ylm (r − R i ). (1.19) Here φinl (|r −R i |) is a radial wave function for an atom located at R i , the subscripts n and l denote the principal quantum number and the azimuthal quantum number,

1.2 Basic Hamiltonian

5

respectively. Ylm (r − R i ) is the spherical harmonics located at atom R i , and m denotes the magnetic quantum number. In the application to solids, it is convenient to make use of real functions {Plν } instead of complex basis functions {Ylm }. The former called the cubic harmonics are constructed by linear combinations of spherical harmonics. Defining the orbital L = (l, ν), the atomic functions for solids are written as follows.

ϕinL (r) = φinl |r − R i | Plν (r − R i ).

(1.20)

The cubic harmonics are constructed to yield the irreducible representation for the cubic symmetry of the point group in crystal. The s (l = 0) function belonging to the A1g representation for the cubic point symmetry is a constant function. 1 Pa1g (r) = Y00 = √ . 4π

(1.21)

The p (l = 1) functions belonging to the T1u representation are defined as follows.

3 x 1 , Pt1u α (r) = √ (−Y11 + Y1,−1 ) = 4π r 2

3 y i , Pt1u β (r) = √ (Y11 + Y1,−1 ) = 4π r 2

3 z . Pt1u γ (r) = Y10 = 4π r

(1.22) (1.23) (1.24)

The d (l = 2) orbitals form the Eg (dγ ) and T2g (dε) representations. The d functions belonging to the Eg representation are given by

5 1 3z2 − r 2 , 4π 2 r2

√ 5 3 x2 − y2 1 . Pegv (r) = √ (Y22 + Y2,−2 ) = 4π 2 r2 2

Pegu (r) = Y20 =

(1.25) (1.26)

The remaining d (l = 2) functions belonging to the T2g representation are given by i Pt2gξ (r) = √ (Y21 + Y2,−1 ) = 2



5 √ yz 3 2, 4π r

5 √ zx 3 2, 4π r

5 √ xy i 3 2. Pt2gζ (r) = √ (−Y22 + Y2,−2 ) = 4π r 2

1 Pt2gη (r) = √ (−Y21 + Y2,−1 ) = 2

(1.27) (1.28) (1.29)

6

1

Introduction to Magnetism

Fig. 1.1 Atomic orbitals from s to d symmetry. The sign of wave functions is shown by + and − in the figure

The angular dependence of the cubic harmonics is shown in Fig. 1.1. The spatial distribution of the atomic wave functions governs the electron hoppings in solids, thus determining the electronic and magnetic properties of solids.

1.3 Formation of Atomic Moments The electronic structure and related properties of an atom may be understood by an independent-electron picture. The spin–orbit interaction in the Hamiltonian (1.13) is smaller than the Coulomb interaction in the ‘light’ elements such as the 3d transition metals. We first neglect the former, and treat the latter by means of the following approximation.  †    † † † aiσ aj σ  akσ  alσ = aiσ alσ aj†σ  akσ  + aiσ alσ aj†σ  akσ   †    † − aiσ akσ  aj†σ  alσ − aiσ akσ  aj†σ  alσ  † †  − aiσ aj σ  akσ  alσ .

(1.30)

Here the average   is taken with respect to an independent particle system which will be chosen later. Note that the two particle operator has been decoupled so that (1.30) exactly holds true when we take the average. This is called the Hartree–Fock approximation. In the Hartree–Fock approximation, the Hamiltonian (1.13) is reduced to H=

 ij σ

† hij σ aiσ aj σ −

   † 1  (Vij kl − Vij lk δσ σ  ) aiσ alσ aj†σ  akσ  . (1.31) 2  ij klσ σ

1.3 Formation of Atomic Moments

7

Here hij σ is given as follows.     n(r  ) 1 † ϕj σ (r) hij σ = ϕiσ (r) − ∇ 2 + vN (r) + dr  2 |r − r  | 

ψ † (r  )ψσ (r) † ϕj σ r  . (r) σ − dr dr  ϕiσ  |r − r |

(1.32)

The Hartree–Fock one-electron wave functions {ϕiσ (r)} are determined by solving the following self-consistent equations.     1 2  n(r ) − ∇ + vN (r) + dr ϕiσ (r) 2 |r − r  |   ϕj∗σ (r  )ϕiσ (r  ) ϕj σ (r) = εiσ ϕiσ (r). nj σ  dr  (1.33) − |r − r  | j

Here εiσ is the Hartree–Fock energy eigen value. When we apply the Hartree–Fock independent particle Hamiltonian in the  average  , nj σ  is given by the Fermi distribution function f (εj σ ) and n(r) = j σ f (εj σ )ϕj∗σ (r)ϕj σ (r) denotes a charge density in the Hartree–Fock approximation. The third term at the l.h.s. of (1.33) gives the electrostatic potential due to electrons. The last term also originates in the Coulomb interaction, but the wave functions have been exchanged due to the anti-symmetric property of the Slater determinant. It is referred as the exchange potential. The Hartree–Fock wave function is the best Slater determinant at the ground state according to the variational principle [4]. Taking the Hartree–Fock wavefunctions as the basis functions, one can express the Hartree–Fock Hamiltonian (1.31) as follows. H=

 iσ

εiσ niσ −

1  (Vijj i − Vij ij δσ σ  )niσ nj σ  . 2 

(1.34)

ij σ σ

Here Vijj i (Vij ij ) is known as the Coulomb integral (exchange integral). (Note that Vij kl should more precisely be Viσj σ  kσ  lσ .) The second term at the r.h.s. is to eliminate the double counting of Coulomb interaction in the Hartree–Fock one-electron energy. The exchange potential in the Hartree–Fock equation (1.33) is nonlocal. Slater proposed an approximate local exchange potential, using the free-electron wave functions. It is given as follows [4]. 

3 vexσ (r) = −3 4π

1/3 nσ (r)1/3 ,

(1.35)

where nσ (r) is the electron density with spin σ . The effective potential in the Hartree–Fock self-consistent equation is spherical in the atomic system when we apply the potential (1.35). We can express the atomic

8

1

Introduction to Magnetism

wave function for an electron as ϕnlmσ (r) = φnlσ (r)Ylm (ˆr ).

(1.36)

Here φnlσ (r) is the radial wave function, and Ylm (ˆr ) is the spherical harmonics. n, l, and m denote the principal quantum number, the azimuthal quantum number, and the magnetic quantum number, respectively. Orbitals l = 0, 1, 2, 3, . . . are called s, p, d, f, . . . , respectively. The nl shells are therefore written as ns, np, nd, nf, etc. In each shell, there are 2l + 1 degenerate orbitals. With the use of the Hartree–Fock atomic orbitals, a many electron state of an atom is expressed in the form of the Slater determinant. The ground-state electronic structure of an atom is constructed according to the Pauli principle. For example, in the case of Ar, we have the ground state 1s2 2s2 2p6 3s2 3p6 . Since the outermost electron shell of Ar is closed, the magnetic moment of Ar vanishes. For an atom with an unfilled shell, the magnetic moment may appear. Since the total number of electrons (N ), the total spin (S), and the total orbital moment (L) of an atom commute with the Hamiltonian, the eigen function Ψ should be specified by the set (N LMSMs ): Ψ (NLMSMs ), where L (S) and M (Ms ) denote the magnitude and z component of orbital moment L (spin S), respectively. The eigen energy, on the other hand, should depend only on N , L, and S because of the rotational symmetry: EA (N LS). The ground state of electrons in an atom should be obtained by minimizing the Coulomb energy. The Coulomb interaction of the Hamiltonian for the unfilled shell is written as follows according to (1.13).     1 Uνν  − Jνν  nν nν  − 2 HCoulomb = Uνν nν↑ nν↓ + Jνν  s ν · s ν  . 2   ν ν>ν

ν>ν

(1.37) Here ν denotes an orbital lm (m = −l . . . l). nν (s ν ) denotes the charge (spin) density operator for electrons in the orbital ν. We have taken into account the Hartree–Fock Coulomb and exchange interactions, Uνν  = Vνν  ν  ν and Jνν  = Vνν  νν  , and omitted the other types of interactions. The first term at the r.h.s. of (1.37) denotes the intraorbital Coulomb interaction, the second term expresses the inter-orbital interaction, and the third term the exchange interaction between the spins on different orbitals. Note that Uνν > Uνν  (ν = ν  ) > Jνν  > 0. Typical values of Coulomb interactions are Uνν ≈ 20 eV, Uνν − Uνν  ≈ 2 eV, and Jνν  ≈ 0.9 eV for 3d transition-metal elements [5]. Note that N stands for the total number of electrons in the unfilled shell when the Hamiltonian (1.37) is applied. The intra-orbital Coulomb interaction (i.e., the first term at the r.h.s. of (1.37)) acts to reduce the doubly occupied states on the same orbital so that it creates active spins on the orbitals of an unfilled shell. The third term in (1.37) aligns the active spins on the orbitals because Jνν  > 0. These effects suggest that the magnitude of the total spin S is maximized at the ground state. This is known as Hund’s first rule, and therefore Jνν  are often called the Hund-rule couplings. On the other hand, the

1.3 Formation of Atomic Moments

9

second term in (1.37) is the energy associated with the configuration of electrons on different orbitals. One might expect that such an interaction energy is reduced when electrons move around the nucleus in the same direction avoiding each other. Thus we expect that the magnitude of the total angular momentum L should be maximized at the ground state under the maximum magnitude of total spin. This is referred as Hund’s second rule. The first and second Hund rules are verified by the full Hartree–Fock numerical calculations. The maximum S and L at the ground state are obtained by using the Pauli principle and the Hund rule, especially from the conditions S = Ms (Max) and L = M(Max) under the maximum S that corresponds to the state Ψ (N LLSS): ⎧N ⎪ for N ≤ 2l + 1, ⎨ S= 2 ⎪ ⎩ 4l + 2 − N for N > 2l + 1, 2  1 N (2l + 1 − N ) for N ≤ 2l + 1, L = 21 2 (N − 2l − 1)(4l + 2 − N ) for N > 2l + 1.

(1.38)

(1.39)

The ground state Ψ (NLMSMs ) is (2L + 1)(2S + 1)-fold degenerate. The multiplet is written as 2S+1 LJ where L takes S, P, D, F, G, H, . . . according to L = 0, 1, 2, 3, 4, 5, . . . , and J (= |L ± S|) is the total angular momentum. These properties are summarized in Fig. 1.2 for 3d transition metal atoms. The value of J at the ground state is determined by the spin–orbit interaction. The spin–orbit interaction (1.16) in the atomic system can be written as HSO =



† ζnl (l)νν  · (s)αγ aνα aν  γ .

(1.40)

 φnl (r)2 dV r dr > 0. dr

(1.41)

ναν  γ

Here 1 ζnl = 2



∞ 0

Assume that the ground state is determined by the Hund rule, and the temperature is such that the corresponding thermal energy is much lower than that of the first excited state of the Coulomb interaction, i.e., the Hund-rule coupling JH : kB T  JH . Here JH is an average value of {Jνν  }, and kB denotes Boltzmann’s constant. Moreover the spin–orbit interaction energy is much smaller than JH in 3d transition metals. In this case, we can neglect all the excited state of the Coulomb interactions, and limit the states to the (2L + 1)(2S + 1) dimensional Hund-rule subspace: {Ψ (NLMSMs )}. We can then verify the following relation in the subspace.  

  Ψ (NLMSMs )HSO Ψ N LM  SMs   

 = λ(NLS) Ψ (N LMSMs )L · S Ψ N LM  SMs . (1.42)

10

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Introduction to Magnetism

Fig. 1.2 The ground state of atoms with 3d unfilled shell. N and mz denote the d electron number and the orbital magnetic quantum number, respectively. Electron configuration for each N in the upper row of the figure shows the state Ψ (NLLSS) leading to the total spin S and the orbital angular momentum L at the ground state. J denotes the total angular momentum. The ground state multiplets (GS) are expressed as 2S+1 LJ . The experimental values of the spin–orbit coupling constant λ are given. Examples of the 3d transition metal ions are also given in the bottom row

Thus we have an effective Hamiltonian for the spin–orbit interaction in the Hundrule subspace as follows. HSO = λ(NLS)L · S.

(1.43)

The coefficient λ(NLS) is obtained by comparing the diagonal matrix element of HSO for the state Ψ (N LLSS) obtained from (1.40) with that of the r.h.s. of (1.43). ⎧ ζnl ⎪ ⎨ for N ≤ 2l + 1, (1.44) λ(NLS) = N ζnl ⎪ ⎩− for N > 2l + 1. 4l + 2 − N Because ζnl > 0, λ(NLS) > 0 for N ≤ 2l + 1 and λ(N LS) < 0 for N > 2l + 1. Thus, J = |L − S| is realized for N ≤ 2l + 1, and J = |L + S| for N > 2l + 1 at the ground state among possible states J = |L + S|, |L + S − 1|, . . . , |L − S|. Therefore J = |L − S| is the ground-state in the light transition-metal elements and the light

1.4 Metal and Insulator in Solids

11

rare-earth elements, and J = |L + S| is the ground state in the heavy transitionmetal elements and the heavy rare-earth elements. Note that the typical value of the spin–orbit coupling is only one tenth of the Hund rule coupling (JH ∼ 1 eV) in the case of 3d transition metal ions as shown in Fig. 1.2. When the spin–orbit coupling is significant as in the rare-earth atoms, the ground-state should be (2J + 1)-fold degenerate instead of (2S + 1)(2L + 1) fold. Ions such as Ti2+ , V3+ , and Cr4+ , for example, have two electrons in the d shell (see the N = 2 column in Fig. 1.2). The Hund rule tells us that the total spin and orbital angular momenta at the ground state are S = 1 and L = 3, respectively. Because the spin–orbit coupling λ > 0 in the light atoms, the total angular momentum at the ground state is given by J = 3 − 1. Thus we have the ground-state multiplet 3 F according to the notation 2S+1 L . In the same way, the ion Co2+ has 7 elec2 J trons in the d shell. According to the Hund rule, we find the ground-state spin and orbital momenta S = 3/2 and L = 3, respectively. Because the spin–orbit coupling λ < 0 in the heavy atoms, the total angular momentum is given by J = 9/2. Thus we obtain the ground-state multiplet 4 F9/2 for Co2+ ion. When the external magnetic field H is applied to the atoms, the energy due to the magnetic field is given by −gJ μB Jz H . Here gJ is Landé’s g factor and is given by gJ = 3/2 + [S(S + 1) − L(L + 1)]/2J (J + 1). The susceptibility which is defined by the induced magnetization divided by H is then given by χ=

gJ2 μ2B J (J + 1) . 3kB T

(1.45)

The above temperature dependence is referred as the Curie law, and C ≡ gJ2 μ2B J (J + 1)/3kB is called the Curie constant. The observation of the Curie law in the susceptibility measurement is regarded as an indication of the existence of local magnetic moments.

1.4 Metal and Insulator in Solids When atoms form a solid, electrons in the outermost shell of atoms start to hop from atom to atom via overlap between atomic orbitals. The question then is whether electrons tend to remain in each atom or to itinerate in the solid. In the case of the former, atomic magnetic moments are well defined in solids and their properties are expected to be explained from the atomic point of view. In the case of the latter, magnetic properties may be quite different from those expected from atomic ones and one has to start from the itinerant limit in order to explain their magnetism. Whether electrons in solids are movable or not is governed by the detailed balance between the energy gain due to electron hopping and the loss of Coulomb interaction. The simplest example may be the case of the hydrogen molecule H2 . According to the molecular orbital picture, the 1s atomic orbitals split into the bonding state with energy ε0 − |t| and the anti-bonding state with energy ε0 + |t| when hydrogen atoms form a molecule. Here ε0 denotes the atomic ground state, and t is the

12

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Introduction to Magnetism

electron hopping integral which is defined by atomic potential v(r) and wave function ϕ(r) as t = ϕ(r − R)v(r − R)ϕ(r) dr. The bonding orbital is occupied by two electrons according to the Pauli principle. It leads to the total energy gain 2|t|. This is the covalent bond in the hydrogen molecule. In this state, electrons hop from one atom to another, and thus they are movable. The covalent bond is stabilized by the kinetic energy gain of independent electrons. It consists of the polarized state in which the 1s orbital of an atom is doubly occupied and the orbital of another atom is unoccupied, and the neutral state in which each atomic orbital is occupied by an electron. When the Coulomb interaction between electrons are taken into account, the covalent bonding state is not necessarily stable because it contains the polarization state with double occupancy on an atomic orbital. Assume that the loss of the intraatomic Coulomb interaction energy in the covalent bonding state is given by U , and consider the neutral-atom state as the state in which each electron is localized on an atom. The total energy in the covalent bonding state is given by E(covalent) = 2ε0 − 2|t| + U , while the energy in the neutral atom state is given by E(neutral) = 2ε0 . Thus, the neutral atom state is realized when the following condition is satisfied. 2|t| < U.

(1.46)

This condition is satisfied when the interatomic distance goes to infinity because |t| goes to zero. Thus the neutral atom state in which electrons are localized on each atom and each atom has a well defined local magnetic moment s = 1/2 is realized in the atomic limit. In solids, we expect the same behavior as found in the hydrogen molecule. Let us consider the behavior of electrons when atoms form a solid. Electrons in solids move in a potential obtained by a superposition of the atomic potentials i v(r − R i ). Here v(r − R i ) denotes the atomic potential on site i. When we adopt the atomic orbitals {ϕiν (r − R i )} as the basis functions and assume the orthogonality between the orbitals, the Hamiltonian for electrons on the outermost shells in solids may be obtained from (1.13) as H = H0 + HCoulomb ,   † εiν niν + tiνj ν  aiνσ aj ν  σ , H0 = iν

HCoulomb =

 iν

−2

Uiνν niν↑ niν↓ +  i

iνj ν 

  i ν>ν 

Jiνν  s iν · s iν  .

(1.47) (1.48)

 1 Uiνν  − Jiνν  niν niν  2 (1.49)

ν>ν 

Here εiν is the atomic level for the orbital ν of site i, and tiνj ν  the transfer integral between the orbital ν at site i and the orbital ν  at site j . The latter is expressed in

1.4 Metal and Insulator in Solids

13

the two-center approximation as  ∗ tiνj ν  = ϕiν (r − R i )v(r − R i )ϕiν  (r − R j ) dr.

(1.50)

Note that we have taken into account in HCoulomb only the intra-atomic Coulomb interactions and omitted the inter-site Coulomb interaction contributions for simplicity. It is not easy to treat the electrons in solids described by the Hamiltonian (1.47) because both the electron hoppings and the Coulomb repulsions have to be taken into consideration. In order to discuss both the itinerant and localized behaviors of electrons in solids, we can consider a simpler Hamiltonian consisting of one orbital per site as follows.    † H= ε0 niσ + tij aiσ aj σ + U ni↑ ni↓ . (1.51) iσ

ij σ

i

Here ε0 , tij , and U denote the atomic level, the transfer integral between sites i and j , and the intra-atomic Coulomb interaction energy, respectively. The Hamiltonian (1.51) is known as the Hubbard model, and was proposed by Gutzwiller and Hubbard independently [6–10]. The Hubbard model (1.51) is the simplest Hamiltonian which describes the motion of interacting electrons in solids. Nevertheless it describes the localization of electrons as well as their itinerant behavior in solids. Let us consider the atomic limit of the model. For an atom, we have 4 atomic states: the empty state (n↑ = 0, n↓ = 0), the single electron states (n↑ = 1, n↓ = 0) and (n↑ = 0, n↓ = 1), and the doubly occupied state (n↑ = 1, n↓ = 1). Associated energies are given as 0, ε0 , ε0 , and 2ε0 + U , respectively. An unfilled shell with spin S = 1/2 appears only for the n = 1 state. Note that there is no Hund’s rule arrangement for spin in this case. In the atomic limit for a solid where tij = 0, electron number ni on eachatom is no longer constant, though the total number of electrons N is given; N = iσ niσ . The eigenstates are given by |Ψ  = |{niσ }, i.e., a set of the electron numbers with spin σ on site i. The eigenenergy for the state is given by

 E {niσ } = (ε0 ni + U ni↑ ni↓ ). (1.52) i

It should be noted that the energy of the system increases by U when the number of doubly occupied states D is increased by one. The ground-state energy E0 of the atomic limit is obtained by minimizing the energy with respect to the number of double occupancy in solid. Assume that the number of lattice points is given by L. When N < L, the ground-state energy is obtained as E0 = ε0 N by choosing D = 0. Magnetic moments on the sites with an electron are active in this case as shown in Fig. 1.3. Because there are L!/N!(L − N )! electron configurations on the L lattice points, the ground state is [2N L!/N!(L − N )!]-fold degenerate.

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Introduction to Magnetism

Fig. 1.3 Electron configurations for less than half filling (upper figure) and for more than half filling (lower figure) in the atomic limit

At half filling, all the atoms are occupied by an electron so that spin degrees of freedom by 2N remain; the degenerated wave functions are given by |1s1z 1s2z 1s3z . . .  when the wave function |{niσ } is written as |n1 s1z n2 s2z n3 s3z . . .  by using the charge ni = ni↑ + ni↓ and the spin siz = (ni↑ − ni↓ )/2. When the electron number N is larger than L, it is no longer possible to keep D = 0; the minimum value of D is given by D = N − L. The ground state energy is then given by E0 = ε0 N +U (N −L). The ground state is [22L−N L!/(2L−N )!(N − L)!]-fold degenerate because there are L!/(2L − N )!(N − L)! configurations for choosing 2L − N(< L) sites with the single electron from L lattice sites and there are 22L−N spin degrees of freedom for each configuration. Note that the spins on 2L − N sites are active in this case. Let us consider the case that electron hopping tij is small but finite. When N = L, electrons are mobile at T = 0 because electrons can move from site to site without increasing the double occupation number D. Thus the system is a metal. However, at half-filling, electron hopping creates a doubly occupied state, thus creating a finite excitation energy by the amount of Coulomb interaction energy U . Therefore electrons cannot move under the infinitesimal electric field. We have then an insulator with local magnetic moments at each site in this case. When there is no Coulomb interaction (U = 0), on the other hand, electrons are generally itinerant. The Hamiltonian is given as H=

 † (H 0 )ij aiσ aj σ .

(1.53)

ij σ

Here (H 0 )ij = ε0 δij + tij (1 − δij ). The noninteracting Hamiltonian is diagonalized   † † by a unitary transformation aiσ = k akσ i|k (aiσ = k akσ i|k∗ ) so that H=



εk nkσ ,

(1.54)

kσ

 where εk = ij k|i(H 0 )ij j |k is an eigen value for the tight-binding one-electron Hamiltonian matrix H 0 , and the set {j |k} (j = 1, . . . , L) is the eigen vector for εk .

1.4 Metal and Insulator in Solids

15

The eigen states for noninteracting Hamiltonian H are given by |Ψ  = |{nkσ }, i.e., a set of electrons with momentum k and spin σ . The eigenenergy is given by

 E {nkσ } = εk nkσ ,

(1.55)

kσ

where the c-number nkσ takes on the value of 0 or 1. The ground state is obtained by putting electrons on the energy levels from the bottom to the Fermi level εF according to the Pauli principle, ε





k <εF

|φ0  =

† akσ

|0,

(1.56)

kσ

so that the ground-state energy is given by k <εF

ε εk . E0 {nkσ } =

(1.57)

kσ

Alternatively, defining the density of states per atom per spin as ρ(ε) =

1 δ(ε − εk ), L

(1.58)

k

we can express the ground-state energy per atom as  E0 = 2

εF

−∞

ερ(ε) dε.

(1.59)

A non-interacting electron system is in general metallic unless the electrons in the atom form a closed shell. The electrons in such systems are mobile. This is because one can add an electron at the energy level just above the Fermi level by applying infinitesimal electric field. Note that spins of itinerant electrons are also mobile. We may expect that there is a transition from metal to insulator at half filling when the intra-atomic Coulomb interaction is increased. Assume that there is a band for a non-interacting system whose band width is W . The center of the gravity of the noninteracting band is assumed to be located at ε0 . When the Coulomb interaction U is increased, each atom tends to be occupied by one electron, and electron hopping to neighboring sites tends to be suppressed in order to reduce the on-site Coulomb interaction energy. In the strongly correlated region, an electron should have a potential ε0 + U on a site having an opposite-spin electron because of the increment of the Coulomb interaction energy due to double occupation, while an electron has a potential ε0 on an empty site. We then expect one more band with the band width of order of W around ε0 + U . The density of states as excitation spectrum is expected to split into two bands at Uc ∼ W (see Fig. 1.4). The formation

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Introduction to Magnetism

Fig. 1.4 The upper and lower Hubbard bands created by on-site Coulomb interaction U

Fig. 1.5 NaCl type structure in NiO

of a gap at the Fermi level implies the existence of an insulator. The insulating state therefore may be realized by the electron correlations when (1.60)

U > W.

This is Hubbard’s alloy analogy picture to the metal–insulator transition [9, 10]. The metal–insulator transition due to electron correlations as mentioned above is commonly known as the Mott transition. The split bands are named the upper and lower Hubbard bands, respectively. The insulator caused by the electron correlations is referred as the Mott insulator. The concept of the Mott insulator was first proposed by Mott [11]. He considered the case of NiO. NiO has the NaCl structure in which Ni–O chain network is formed along [100] direction (see Fig. 1.5). The electronic configuration of the Ni28 (O8 ) atom is given by 1s2 2s2 2p6 3s2 3p6 3d8 4s2 (1s2 2s2 2p4 ). The oxygen atoms are considered to form a closed shell in the compound taking electrons from Ni atoms, so that we have Ni2+ :

1s2 2s2 2p6 3s2 3p6 3d8

O2− :

1s2 2s2 2p6 .

In this case, the Fermi level should be on the d bands in the crystalline system, and we can expect a metal because the 5-fold d bands overlap each other in general (see Fig. 1.6).

1.4 Metal and Insulator in Solids

17

Fig. 1.6 Density of states for NiO in the paramagnetic state obtained by the band calculation [12]. The vertical dashed line shows the Fermi level

Experimental results however indicate that NiO is an insulator. To clarify the insulating behavior, Mott considered the electron hopping on the NiO network. Let us estimate an excitation energy Eg when an electron moves to the neighboring Ni2+ site. This process is similar to an excitation of the H2 molecule to a polarized state. 2+ 2−

Ni O 2 −→ Ni3+ O2− + Ni+ O2− .   Assume that the intra-atomic Coulomb interaction is given by i (νσ,ν  σ  ) U · niνσ niν  σ  for simplicity. There an electron is coupled to the remaining 7 electrons via the on-site Coulomb interaction. After the hopping, the electron is coupled to 8 electrons at the neighboring site, so that the Coulomb interaction is increased by U . On the other hand, the kinetic energy gain due to electron hopping at Ni3+ and Ni2+ sites may be given by 2z|t| ∼ W,

(1.61)

because electrons in surrounding Ni sites can enter into the Ni3+ ions and the electrons in Ni2+ ions can jump to the surrounding Ni sites. Here z is the number of Ni nearest neighbors, and t is an effective electron hopping between Ni atoms. Note that |t| corresponds to an energy gain for the formation of the bonding state due to electron hopping in the H2 molecule, while U corresponds to the Coulomb energy loss in the polarized state. The excitation energy of an electron hopping is therefore given by Eg ∼ U − W.

(1.62)

Thus, we again have the same criterion for the formation of the Mott insulator (1.60) from the condition Eg > 0. The magnetic materials in the insulator are considered to be the Mott-type insulator in which the atomic magnetic moments are built up in the unfilled shell. CrO2 and CrBr2 are typical ferromagnetic insulators, while MnO, FeO, CoO, and NiO

18

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Introduction to Magnetism

are known as antiferromagnetic insulators in which the atomic magnetic moments change their direction alternatively from site to site. On the other hand, there are many magnetic metals where the polarized electrons are mobile. These materials are referred as the metallic magnets or itinerant electron magnets. The magnetic properties of these systems are known as the metallic magnetism or itinerant magnetism. Fe, Co, and Ni are typical examples of the metallic ferromagnet, while Cr and Mn alloys show the antiferromagnetism. We describe in this book the itinerant magnetism of metals and alloys.

1.5 Quenching of Orbital Magnetic Moments The expectation value of the total orbital magnetic moment M L  = L is usually quenched in the crystalline system due to the crystalline potential of electrons. Let us consider the case where the spin–orbit interactions are negligible as in the 3d transition metals, and that the magnetic field is not applied. Moreover we may assume in solids that the ground state is nondegenerate. One can then choose the ground-state wave function to be real; Ψ ∗ = Ψ . The angular momentum L is real because it is a physical quantity: L = L∗ .

(1.63)

Using the relation Ψ ∗ = Ψ , the r.h.s. of the above equation is written as L∗ = −Ψ |L|Ψ .

(1.64)

L = 0.

(1.65)

Therefore (1.63) indicates that

Thus, one can expect in metals with sufficiently small spin–orbit interactions that only spin magnetic moments contribute to the magnetization. M = 2S.

(1.66)

This is referred as the quenching of orbital magnetic moments. In the atomic system, electrons are bound by the nuclei, and the orbital angular momentum is finite in general according to Hund’s second rule. In the crystalline system, the Coulomb energy gain leading to Hund’s second rule is not expected because electrons can hop to the neighboring sites to reduce the Coulomb energy, and electrons can move turning around to right and left (see Fig. 1.7). It may lead to L = 0. This is a physical reason for the quenching of orbital moments in solids. In the 3d transition metals and alloys which we will discuss, the orbital magnetic moments are usually quenched. In the system with a large spin–orbit coupling such as the rare-earth metals or the system with a strong magnetic field, the orbital moments remain.

1.6 Heisenberg Model

19

Fig. 1.7 Concept of orbital magnetic moments in atom (L = 0: left) and solids (L = 0: right)

1.6 Heisenberg Model In the insulator, magnetic behaviors are often described by a simple Hamiltonian known as the Heisenberg model. We briefly discuss the Hamiltonian and related properties of the insulator magnetism in this section. Let us consider the hydrogen molecule. We assume one orbital for each hydrogen atom, ϕ1 for atom 1 and ϕ2 for atom 2, and neglect the overlap integrals for simplicity, i.e., ϕ1 |ϕ2  = 0. The Hamiltonian for the hydrogen molecule is obtained from (1.13) as we derived (1.47).      1 † H= ε0 niσ + tij aiσ aj σ + U ni↑ ni↓ + K − J n1 n2 − 2J s 1 · s 2 . 2 iσ

ij σ

i

(1.67) Here K and J denote the intersite Coulomb and exchange integrals given by  K=  J=

dr dr 

ϕ1∗ (r)ϕ2∗ (r  )ϕ2 (r  )ϕ1 (r) , |r − r  |

(1.68)

dr dr 

ϕ1∗ (r)ϕ2∗ (r  )ϕ1 (r  )ϕ2 (r) . |r − r  |

(1.69)

In the atomic subspace where n1 = n2 = 1 and the electron hopping is suppressed, the Hamiltonian (1.67) reduces as follows. 1 H = 2ε0 + K − J − 2J s 1 · s 2 . 2

(1.70)

It is diagonalized with the use of the total spin S (in the representation of S and Sz ) as   H = 2ε0 + K − J S(S + 1) − 1 .

(1.71)

Therefore the hydrogen molecule has the energy 2ε0 + K − J for the triplet state and 2ε0 + K + J for the singlet state when the inter-site atomic distance is large. This is the Heitler–London theory for the hydrogen molecule. The last term in (1.70) denotes an interaction between the atomic spins s 1 and s 2 . In general, the following Hamiltonian showing the interactions between atomic

20

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Introduction to Magnetism

spins {S i } is known as the Heisenberg model.  H =− Jij S 1 · S 2 .

(1.72)

(i,j )

The interaction Jij = 2J in the Heitler–London theory is the inter-site exchange integral due to the Coulomb interaction. This coupling in the Heisenberg model is referred as the direct exchange interaction. The direct exchange interaction is generally positive in sign (i.e., ferromagnetic). Another type of the Heisenberg model with effective exchange coupling is possible in the insulator. In order to derive the effective Hamiltonian, we first derive the perturbative effective Hamiltonian from a more general point of view [13]. The effective Hamiltonian is constructed such that it yields the same energy eigen values in a subspace as in the original Hamiltonian acting on the full Hilbert space. Let us consider a subspace  P spanned by a set of states {|s}, and introduce a projection operator P = s |ss| which projects any state onto subspace P. The projection operator which chooses the complementary subspace Q is defined by Q = 1 − P . Note that P 2 = P , Q2 = Q, and P Q = QP = 0. The eigen value problem of the original Hamiltonian H is written as (H − E)ψ = 0.

(1.73)

Inserting the identity 1 = P + Q into the above equation as (P + Q)(H − E)(P + Q)ψ = 0 and multiplying P from the l.h.s. of the equation, we obtain   (P H P − E)P + P H Q ψ = 0. (1.74) In the same way, we can derive the following equation   QH P + (QH Q − E)Q ψ = 0.

(1.75)

In (1.74) and (1.75), the wave function ψ is separated into two components belonging to different subspaces, P ψ and Qψ . From both equations, one can eliminate the wave function Qψ that leads to the eigenvalue equation in subspace P, namely,   P H P − P H Q(QH Q − E)−1 QH P P ψ = EP ψ. (1.76) The above equation indicates that the following Hamiltonian Hp (E) acting on the wave function P ψ in the subspace P is regarded as an effective Hamiltonian which yields the same eigen value E as in the original Hamiltonian H , where Hp (E) = P H P − P H Q(QH Q − E)−1 QH P .

(1.77)

The effective Hamiltonian depends on the energy which should be obtained selfconsistently. Taking the subspace spanned by a subset of the eigenfunctions of an

1.6 Heisenberg Model

21

unperturbed Hamiltonian, one can derive the effective Hamiltonian which is correct up to the second order of the interaction Hamiltonian. Assume that the Hamiltonian consists of a noninteracting part H0 and an interaction part HI such that H = H0 + HI . The eigenvalues {Ek0 } and eigenfunctions {φk } for H0 are assumed to be known. We define the subspace P spanned by a part of {|φk }. The projection op erator is then defined by P = P μ |φμ φμ |. We have the relation QH P = QHI P , and thus obtain the effective Hamiltonian which is correct up to the second order in interaction, after having replaced the Hamiltonian H and the energy E in the denominator of (1.77) with the zeroth-order ones, H0 and Ek0 , i.e.,

−1 Hp = P H P − P HI Q QH0 Q − Ek0 QHI P .

(1.78)

The effective Hamiltonian depends on the eigen state k explicitly. Therefore we rewrite the second term as follows. P  Q 

−1 0

−1 QH0 Q − Ek0 QHI P φk = Eν − Eμ0 |φν φν |HI |φμ φμ |φk . μ

ν

(1.79) Thus the perturbative Hamiltonian (1.78) is expressed as follows.

−1 0 Hp = P H P − P HI Q EQ − EP0 QHI P .

(1.80)

0 ) denotes the eigenvalues belonging to subspace P (Q). The eigen Here EP0 (EQ value equation for the perturbative effective Hamiltonian is then given by

Hp P ψk = Ek P ψk .

(1.81)

We can derive the effective Heisenberg model in the strong Coulomb interaction regime at half-filling using the formula (1.80). Let us consider the Hubbard model (1.51). As found in Sect. 1.4, the ground state of the atomic limit at halffilling is specified by {ni = 1} states. The ground state is 2L -fold degenerate, L being the number of sites. We start from the ground state in the atomic limit at half filling, i.e., the {ni = 1} states, and define the subspace P from them. The complementary subspace Q consists of the subspace containing empty states and the subspace containing the doubly occupied states. In the strongly correlated region, the electron hopping rate |tij | is much smaller than the Coulomb interaction strength U , so that the Hamiltonian H is separated into the atomic state    † aj σ . H0 = iσ ε0 niσ + i U ni↑ ni↓ and the ‘interaction’ term HI = ij σ tij aiσ The first term at the r.h.s. of (1.80) is given by P Lε0 P . In the calculations of  † † the second term, we have QHI P = ij σ tij aiσ aj σ P because aiσ aj σ P belong to 0 − E 0 = U when it is operated on the single doublythe space Q. Moreover, EQ P occupied state QHI P . Thus we obtain the expression for the second term at the r.h.s.

22

1

Introduction to Magnetism

of (1.80) as follows.  tlk tij

−1 0 † † P HI Q EQ − EP0 QHI P = P alσ  akσ  aiσ aj σ P . U 

(1.82)

ij klσ σ

Among the summation at the r.h.s. of (1.82) only the k = i and l = j terms remain. Rearranging the creation and annihilation operators, we obtain the relation.  σσ

1 † aj†σ  aiσ  aiσ aj σ = nj − ni nj − 2s i · s j . 2 

(1.83)

Substituting the above relation into (1.82), we obtain the following effective Hamiltonian.     4|tij |2 1  |tij |2 si · sj P . Hp = P L ε0 − (1.84) + 2 U U j

(i,j )

Apart from the constant term, we arrive at the following effective Heisenberg model in the strong Coulomb interaction regime at half filling, which operates on the subspace P.  Jij s 1 · s 2 . (1.85) H =− (i,j )

Here Jij = −

4|tij |2 . U

(1.86)

The effective exchange integral (1.86) is known as the super exchange interaction because it is caused by a virtual exchange of electrons belonging to different atoms via transfer integrals. The super exchange interaction is antiferromagnetic, while the direct exchange interaction (1.69) is ferromagnetic. Note that the subspace on which the Hamiltonian (1.85) acts is given by {P |n1 s1z , n2 s2z , . . .} = {|1s1z , 1s2z , . . .}. This can be simplified as {|s1z , s2z , . . .}. In many magnetic insulators, anions are often located between the magnetic ions. In such a case, the super exchange interaction becomes dominant. The Heisenberg model (1.85) is not justified in the metallic system where the band width W ∼ U . Nevertheless, we often find the same type of the intersite magnetic interactions, which are useful for qualitative understanding of the magnetism (see Sects. 6.4 and 8.3 for examples).

1.7 Magnetic Structure Magnetic materials have their microscopic structure for arrangement of atomic magnetic moments. This is called the magnetic structure. The structure characterizes the

1.7 Magnetic Structure

23

magnetic property as well as the electronic structure. In this section, we introduce the Fourier representation of the magnetic structure and explain briefly the magnetic structures in solids. The magnetic structure is characterized by the magnetic moment density which is given by   (1.87) M(r) = ψ † (r)mψ(r) . Here ψ(r) = (ψ↑ (r), ψ↓ (r)) is the field operator of electrons. m denotes the magnetic moment for an electron as given by (1.7). In a crystalline system, we may divide the space into the Wigner–Seitz cells for each atom. Then we can define a magnetic moment of atom i as follows.  (1.88) mi = M(r) d 3 x. i

Here the integration is over the Wigner–Seitz cell belonging to atom i. In a noncrystalline system we may adopt the Voronoi polyhedra instead of the Wigner–Seitz cells. In the case of the crystalline system, the atomic position R l is expressed by using the primitive translational vectors a, b, c as R l = l1 a + l2 b + l3 c.

(1.89)

The atomic magnetic moment is expanded with use of the Fourier lattice series as follows.  ml = m(q)eiq·R l . (1.90) q

Here q is the wave vector in the first Brillouin zone. The Fourier components of the magnetic moments are given as m(q) =

1 ml e−iq·Rl , L

(1.91)

l

L being the number of lattice points. Note that m(−q) = m(q)∗ because ml are real. Moreover we assumed that there is only one atom per unit cell. When there are more than one atom per unit cell, we have to add the atomic position in a unit cell ηλ to (1.89). Accordingly, the magnetic moments (1.90) and (1.91) are specified by (λ) the type of atom λ as ml (m(λ) (q)). Hereafter we consider the crystalline system with one magnetic atom per unit cell. The microscopic magnetic structure of magnetic materials is specified by a set of {ml } or {m(q)}. The simplest structure is that all of the atomic magnetic moments have the same direction. It is realized when all the Fourier components vanish except m(q = 0) = m, and is known as the ferromagnetic structure (see Fig. 1.8). Typical transition metals such as Fe, Co, and Ni exhibit the ferromagnetism.

24

1

Introduction to Magnetism

Fig. 1.8 Ferromagnetic arrangement

Fig. 1.9 Antiferromagnetic arrangement

The antiferromagnetic structure (AF) is defined by alternative arrangement of the atomic magnetic moments along a direction, e.g., the z direction (see Fig. 1.9). It is specified by a set, m(±Q) = mk exp (±iα) and m(q = ±Q) = 0, where Q consists of half a reciprocal lattice vectors, and k denotes a unit vector along the z axis, and α denotes a phase of the structure. In the case α = 0, we have ml = mk cos(Q · R l ).

(1.92)

When the position R l moves along Q vector, Q · R l changes by π . Accordingly, the magnetic moment ml changes its sign. The AF structure on the simple cubic lattice as shown in Fig. 1.10(a) is expressed by the wave vector Q = (K 1 + K 2 + K 3 )/2 = (1, 1, 1)π/a, a being the lattice constant. The AF structure on the body-centered cubic lattice as shown in Fig. 1.10(b) is expressed by the wave vector Q = (−K 1 + K 2 + K 3 )/2 = (0, 0, 1)2π/a. In the case of the face-centered cubic lattice (fcc) structure, two kinds of AF structures are known. The AF structure of the first-kind as shown in Fig. 1.10(c) is described by a wave vector Q = (K 2 + K 3 )/2 = (0, 0, 1)2π/a. The atomic moment alternatively changes the direction with a translation by (0, 0, 1)a/2. It is also possible to change direction alternatively along 1 1 1 axis. This is referred as the AF structure of the second-kind, and is characterized by Q = (K 1 + K 2 + K 3 )/2 = (1, 1, 1)π/a (see Fig. 1.11). Cr with 1 at% Mn on the bcc lattice shows the AF structure, and γ -Mn on the fcc lattice shows the AF structure of the first-kind according to the neutron diffraction experiments. It should be noted that the network consisting of the up-spin atoms (or the downspin atoms) forms a lattice referred as the sublattice. There are two sublattices in the antiferromagnetic structures shown in Fig. 1.10. Each sublattice forms the fcc lattice with the lattice constant 2a in the case of the sc structure (Fig. 1.10(a)), the sc lattice with the lattice constant a in the case of the bcc structure (Fig. 1.10(b)), √ and the simple tetragonal structure with the lattice constants a/ 2 and a in the case of the fcc structure (Fig. 1.10(c)), respectively. The sinusoidal spin density wave (SDW) structure is expressed by ml = mk sin(Q · R l + α),

(1.93)

where Q vector is neither equal to 0 nor to the AF wave vectors. The magnetic moment sinusoidally changes with a period λ = 2π/|Q| along the Q direction (see

1.7 Magnetic Structure

25

Fig. 1.10 Antiferromagnetic arrangements on the simple cubic lattice (a), body-centered cubic lattice (b), and face-centered cubic lattice (c). The lattice constants are shown by a Fig. 1.11 The antiferromagnetic structure of the second-kind on the face-centered cubic lattice

Fig. 1.12 Spin density wave structure

Fig. 1.12). Note that the period is not necessarily commensurate with the lattice spacing in general. Cr is a well-known example of the SDW. It is also possible that the magnetic moment rotates with a translation (see Fig. 1.13). This is known as the helical structure, and is expressed as   ml = m e1 cos(Q · R l + α) + e2 sin(Q · R l + α) . (1.94) Here e1 and e2 are the unit vectors being orthogonal to each other. The magnetic moment with amplitude m rotates on the e1 –e2 plane with the translation along the Q vector. The above expression (1.94) can also be written as ml = m(Q)eiQ·R l + m(Q)∗ e−iQ·R l ,

(1.95)

with m(Q) =

m (e1 − ie2 )eiα . 2

(1.96)

It indicates that the helical structure is specified by the Q vector and the condition m(Q)2 = 0.

(1.97)

26

1

Introduction to Magnetism

Fig. 1.13 Helical structure (left) and conical structure (right)

Fig. 1.14 2Q multiplespin-density wave structure

It is known from the neutron experiments that Au2 Mn, MnO2 , CrO2 , Eu, Tb, Dy, Ho etc. show the helical structure. So far the magnetic structures are characterized by only one Q vector. One can also consider the helical type structure with the bulk magnetization as follows.

ml = mz k + m i cos(Q · R l + α) + j sin(Q · R l + α) .

(1.98)

The above expression is alternatively written as ml = mz k + m(Q)eiQ·R l + m(Q)∗ e−iQ·R l ,

(1.99)

m(Q)2 = 0.

(1.100)

with

This is referred as the conical magnetic structure (see Fig. 1.13). The magnetic moment rotates for example on the x–y plane when the moment translates along the direction of Q.

1.7 Magnetic Structure

27

Fig. 1.15 3Q multiplespin-density wave structure

One can also consider a magnetic structure consisting of two wave vectors, Q1 and Q2 . For example, we can consider a structure such as ml = m(Q1 )eiQ1 ·Rl + m(Q2 )eiQ2 ·R l + c. c.

(1.101)

Assuming that m(Q1 ) = mx i/2 and m(Q2 ) = mx j /2, we have ml = mx i cos(Q1 · R l ) + my j cos(Q2 · R l ).

(1.102)

This is known as a double Q multiple SDW (2Q-MSDW) (see Fig. 1.14). In the case of the fcc lattice, we may consider the wave vectors Q1 = (1, 0, 0)2π/a and Q2 = (0, 1, 0)2π/a. Then the x(y) component changes the sign with a translation R l = (1, 0, 0)a/2 (R 2 = (0, 1, 0)a/2) as shown in Fig. 1.14. We can also consider the magnetic structure consisting of three wave vectors Q1 , Q2 and Q3 , which is the so-called triple Q multiple SDW (3Q-MSDW). For example, we have a 3Q-MSDW such that ml = mx i cos(Q1 · R l ) + my j cos(Q2 · R l ) + mz k cos(Q3 · R l ), (1.103) with Q1 = (1, 0, 0)2π/a, Q2 = (0, 1, 0)2π/a, and Q3 = (0, 0, 1)2π/a. Each component changes sign after the translation by a/2 along the same direction as shown in Fig. 1.15. In the substitutional disordered alloys, we have more complicated structures which cannot be described by a small number of wave vectors. When we take a configurational average of the atomic magnetic moments, we can define the average magnetization [ml ]c . Here [∼]c denotes the configurational average. In some

Fig. 1.16 Spin glass arrangement

28

1

Introduction to Magnetism

cases, an ordered state with randomly oriented magnetic moments referred as the spin glass (SG) appears (see Fig. 1.16). The SG is considered to be characterized by the condition such that   [ml ]c = 0 and m2l c = 0. (1.104) The disordered dilute alloys such as Cu1−x Mnx (x  0.1) and Au1−x Fex (x  0.1) are known to show a SG at low temperatures (see Sect. 7.1). Experimentally most magnetic structures of crystalline systems are determined by the neutron elastic scattering experiments. The readers who are interested in the experimental determination of the magnetic structure are recommended to refer to the books by Marshall and Lovesey [14, 15].

Chapter 2

Metallic Magnetism at the Ground State

We deal with the ferromagnetism in metals at the ground state in this chapter. The stability of ferromagnetism is one of the fundamental problems in magnetism. Since the spin polarization is carried out by itinerant electrons in metals, we start from the band model to consider the stability. The Hartree–Fock approximation overestimates the magnetic energy gain due to spin polarization. We clarify the effects of electron correlations on the ferromagnetic instability by using the low density approximation and the Gutzwiller variational theory. These theories lead to the notion of an effective Coulomb interaction. The density functional theory (DFT) provides us with a quantitative description of the ground-state magnetism in metals. We present the band theory of ferromagnetism based on the DFT in the second half of the chapter, and discuss quantitative aspects of the ferromagnetism of 3d transition metals at the ground state.

2.1 Band Theory of Ferromagnetism When atoms form a solid, most of the elements on the periodic table lose the atomic magnetic moments and show a simple paramagnetism. Only a few metals such as Fe, Co, and Ni in 3d transition metal series and heavy rare earth metals from Gd to Tm show the ferromagnetism. In the case of the rare-earth system, the 4f orbitals are well localized so that the atomic magnetic moment on each atom remains. The ferromagnetism of rare-earth metals is realized by the ferromagnetic couplings between these atomic magnetic moments [16]. On the other hand, in the 3d metal system, electrons in the 3d unfilled shell are considered to be itinerant and to form bands near the Fermi level. In fact, the atomic moment model does not explain the experimental observations that the ground-state magnetizations of Fe, Co, and Ni show the non-integer values of 2.2, 1.7, and 0.6 μB per atom, respectively. The Sommerfeld coefficients of specific heat show large values of 5–7 mJ/K2 mol in these ferromagnets, indicating the formation of narrow d bands. The ferromagnetism in 3d transition metals is therefore considered to be caused by the 3d itinerant electrons. Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_2, © Springer-Verlag Berlin Heidelberg 2012

29

30

2

Metallic Magnetism at the Ground State

Fig. 2.1 Spin polarization in bands

A way to clarify the origin of the itinerant magnetism at the ground state is to start from the band limit and to take into account the electron–electron interactions. Let us consider the ferromagnetism in metals, and assume that the itinerant band electrons are polarized. The polarization gives the magnetization per atom as m = n↑  − n↓ . Here nσ  denotes the average number of electrons with spin σ per atom. The spontaneous spin polarization cannot be explained without the Coulomb interaction. Let us assume that the down-spin electrons in a small energy range [εF − Δ, εF ] just below the Fermi level εF move to the up-spin band on the Fermi level as shown in Fig. 2.1. The number of moved electrons δN is given by δN = Δ · Lρ(0), where ρ(0) is the density of states (DOS) per atom and per spin at the Fermi level in the paramagnetic state. L denotes the number of lattice points. Therefore, we have the energy width Δ as Δ = δN/Lρ(0). The change of the band energy E1 is given by E1 = Δ · δN =

(δN )2 > 0. Lρ(0)

(2.1)

Thus the ferromagnetic spin polarization in the noninteracting system always increases the energy irrespective of details of the band structure. One needs the intraatomic Coulomb interaction U to stabilize the ferromagnetism. In order to understand the mechanism, we consider the energy change E2 of Coulomb interactions due to small polarization 2δN . Consider the single band model with the intra-atomic Coulomb interaction U . In a mean-field approximation, the intra-atomic Coulomb interaction energy may be given by L U n↑ n↓  ∼ L U n↑ n↓ .

(2.2)

Then, the Coulomb energy change is given by 

δN E2 ∼ −L U L

2 < 0.

(2.3)

2.1 Band Theory of Ferromagnetism

31

The total energy change E = E1 + E2 is given by   (δN )2 1 −U . E ∼ ρ(0) L

(2.4)

Therefore the ferromagnetism is stabilized when E < 0, i.e., ρ(0) U > 1.

(2.5)

The above condition for the appearance of the ferromagnetism is called the Stoner condition. The ferromagnetism is stabilized when the energy gain of the Coulomb interaction due to small spin polarization overcomes the band energy loss. In the case of the single band model, the spin polarized state can be stabilized by reducing the double occupancy n↑ n↓ . The Stoner criterion and the magnetization are obtained by the Hartree–Fock approximation more clearly. Let us consider first the single band for a qualitative argument, and adopt the Hubbard model (1.51). In the Hartree–Fock approximation, the interaction term is decoupled as

U n↑ n↓ ≈ U n↑ n↓  + n↑ n↓ − n↑ n↓  . (2.6) The approximation neglects the fluctuation term U δn↑ δn↓ . Here δnσ = nσ − nσ . Using the approximation, we obtain the Hartree–Fock Hamiltonian as follows.   

† H˜ = ε0 + U ni−σ  niσ + tij aiσ aj σ − U ni↑ ni↓ . (2.7) iσ

ij σ

i

Here U ni−σ  is the Hartree–Fock potential for an electron with spin σ . The total energy in the Hartree–Fock approximation is given by     † E= ε0 ni  + tij aiσ aj σ + U ni↑ ni↓ . i

ij σ

(2.8)

i

At the ground state, the energy is written as  E= εk nkσ  + L U n↑ n↓ .

(2.9)

k

Here εk is the band energy obtained by diagonalizing the noninteracting Hamiltonian matrix (H 0 )ij = ε0 δij + tij (1 − δij ). nkσ is the electron  occupation number for an electron with momentum k and spin σ , and nσ  = kσ nkσ /L. The first (second) term at the r.h.s. of (2.9) is the kinetic energy E1 (interaction energy E2 ) discussed in (2.1) and (2.3). When the potential is spatially uniform, one can obtain a self-consistent equation for the average electron number using the Hartree–Fock Hamiltonian (2.7) as follows. 

niσ  = dω f (ω)ρ 0 ω − U ni−σ  + μ . (2.10)

32

2

Metallic Magnetism at the Ground State

Here f (ω) = 1/(eβω + 1) is the Fermi distribution function, β being the inverse temperature. ρ 0 (ω) is the DOS per atom per spin for the noninteracting Hamiltonian, and μ denotes the Fermi level.  The above equations are expressed by the local charge n = σ niσ  and mag netization m = σ σ niσ  as follows.   1 1 dω f (ω)ρ 0 ω − U n + U mσ + μ , 2 2 σ     1 1 0 m = σ dω f (ω)ρ ω − U n + U mσ + μ . 2 2 σ

n =



(2.11) (2.12)

The first equation determines the Fermi level μ when the electron number n is given, while the second one determines the magnetization m self-consistently. In the same way one can also express the Hartree–Fock energy per atom as follows.     1 1 E/L = μn + dω f (ω)ω ρ 0 ω − U n + U mσ + μ 2 2 σ

1 − U n2 − m2 . 4

(2.13)

Here we omitted the site suffix for simplicity assuming uniform charge and spin polarization. Appearance of the ferromagnetism may be characterized by an instability of the paramagnetic state. The latter is obtained from the uniform susceptibility. Let us apply an infinitesimal magnetic field h. We then have additional potential −hσ in the Hartree–Fock self-consistent equations. The uniform paramagnetic spin susceptibility χ is given by χ = [∂m/∂h]h=0 . Using (2.12) with additional potential −hσ , we obtain the Hartree–Fock susceptibility at the ground state as follows. χ=

2ρ(0) . 1 − ρ(0)U

(2.14)

Here ρ(0) is the DOS per atom per spin at the Fermi level in the nonmagnetic state. When ρ(0)U = 1, the susceptibility diverges, and we again obtain the Stoner condition (2.5). In order to obtain the magnetization we have to solve the self-consistent equations (2.11) and (2.12). There are two types of the ferromagnetism in general. When the Coulomb interaction strength is large enough, we can expect that the up-spin band is below the Fermi level so that the magnetization is not changed by the magnetic field. Another case is that holes remain in the up-spin band so that both the up and down bands are laid on the Fermi level as shown in Fig. 2.2. The former is called the complete ferromagnetism or the strong ferromagnetism, while the latter is called the incomplete ferromagnetism or the weak ferromagnetism.

2.1 Band Theory of Ferromagnetism

33

Fig. 2.2 The strong (left) and weak (right) ferromagnets

To describe the ferromagnetism in 3d transition metals in more detail, we have to take into account five d orbitals. The Hartree–Fock approximation to the degenerateband Hamiltonian (1.47) yields the following effective Hamiltonian (see (1.34)). H˜ =



εiLσ niLσ +

iLj L

iLσ

H1  =

 1 



† tiLj L aiLσ aj L σ − H1 ,

  1  1 Umm niL  + Umm − Jmm niL niL  4 m 2  2 mm 1 1  − Umm miL 2 − Jmm miL miL  . 4 m 4 

i

(2.15)

2

(2.16)

mm

Here L (L ) denotes the atomic orbital lm (lm ) with l = 2. εiLσ is the atomic level with the Hartree–Fock potential, and is given by εiLσ = εiL − εiL = εL0

   1 Umm miL  + Jmm miL  σ, 2 

1 − μ + Umm niL  + 2

 m =m

(2.17)

m =m

 1 Umm − Jmm niL . 2

(2.18)

Note that εL0 is the atomic level and we have introduced the chemical potential for convenience. The local charge and magnetization for orbital L are given by nL  =

 nLσ ,

(2.19)

σ

mL  =

 σ

σ nLσ ,

(2.20)

34

2

Metallic Magnetism at the Ground State

and  nLσ  =

dω f (ω)ρLσ (ω).

(2.21)

We have omitted the site indices i in the above expressions for simplicity. ρLσ (ω) is the Hartree–Fock DOS for orbital L and spin σ . ρLσ (ω) =

  iL|k2 δ(ω − ε˜ kσ ).

(2.22)

k

˜ 0 )iLj L = εLσ δij δLL + Here ε˜ kσ is the eigen value for the Hamiltonian matrix (H tiLj L , and iL|k is the overlap integral between the local orbital iL and the oneelectron eigen state k. For simplicity, we neglect in the following the orbital dependence of the Coulomb and exchange integrals as Umm = U0 δmm + U1 (1 − δmm ),

(2.23)

Jmm = J (1 − δmm ).

(2.24)

Here U0 (U1 ) is the average intraorbital (interorbital) Coulomb interaction and J is the average exchange interaction. Moreover we assume that the orbitals form a basis set of the irreducible representation of point symmetry. We have then

 1 U0 mΓ  + J m − mΓ  σ, 2  

1 1 0 εL = εL − μ + U0 nΓ  + U1 − J n − nΓ  . 2 2 εiLσ = εL −

(2.25) (2.26)

Here nΓ  (mΓ ) denotes the average electron number of orbital L, i.e., nL  (magnetic moment mL ) belonging   to the point symmetry representation Γ . Moreover, n = Γ dΓ nΓ  (m = Γ dΓ mΓ ) denotes the total charge (magnetization) on a site, and dΓ is the number of orbitals belonging to the point symmetry Γ . When all the magnetic moments on each orbitalare assumed to make the same contribution, we have mΓ  = m/D, with D = Γ dΓ being the number of orbital degeneracy. In this case, the atomic level is given by 1 εLσ = εL − J˜mσ − hσ, 2

(2.27)

  1 1 ˜ J. J = U0 + 1 − D D

(2.28)

and

2.1 Band Theory of Ferromagnetism

35

We added the magnetic field h for convenience. The average electron number is then given by    1 ˜ nLσ  = dω f (ω)ρL ω + J mσ + hσ . (2.29) 2 Here ρL (ω) is the DOS for orbital L per spin in the nonmagnetic state. Taking the same step as in the single band case, we obtain the uniform susceptibility as χ=

2ρ(0) 1 − ρ(0)J˜

.

(2.30)

 Here ρ(0) = Γ dΓ ρΓ (0) is the DOS per atom per spin, and J˜ is the effective exchange interaction given by (2.28). When there is only one symmetry band Γ on the Fermi level and the other bands are below it, we have the atomic level (2.27) in which the effective exchange interaction J˜ has been replaced by   1 1 ˜ J= J. (2.31) U0 + 1 − dΓ dΓ Therefore we have again the same form of susceptibility (2.30). The susceptibility (2.30) indicates that the Stoner condition to the degenerateband system is given by ρ(0) J˜ > 1.

(2.32)

As seen from (2.28) and (2.31), the intraorbital Coulomb interaction U0 which is much larger than J remains in the effective exchange interaction J˜ even in the degenerate bands system in the Hartree–Fock approximation. According to the Hartree–Fock atomic calculations, the intra- and inter-orbital Coulomb interactions in Fe are U0 = 25 eV and J = 0.9 eV, respectively [5]. Using (2.28), we obtain J˜ = 5.8 eV. The band width of Fe is estimated from a band calculation as W ≈ 5 eV [17], and consequently we expect ρ(0) ∼ 1 assuming a rectangular DOS. Therefore we expect that ρ(0)J˜ ∼ 6, and the Stoner condition (2.32) explains the ferromagnetism of Fe. The same arguments are also possible for Ni. There we have U0 = 28 eV and J = 1.0 eV, respectively. Using (2.31) in this case, we obtain J˜ = 10 eV. The band width of Ni is estimated to be W ≈ 4 eV, and thus ρ(0) ∼ 0.75 assuming a rectangular DOS consisting of the t2g bands at the Fermi level. We have then ρ(0)J˜ ∼ 10, which is consistent with the ferromagnetism in Ni. However the Stoner condition in the Hartree–Fock approximation (2.32) also predicts the ferromagnetism for the nonmagnetic transition metals. For example, in the case of Ti we have U0 = 18 eV and J = 0.6 eV, and thus we obtain J˜ = 10 eV using (2.28). The band width of Ti is estimated to be W ≈ 5.5 eV, and thus ρ(0) ∼ 0.9. We have then ρ(0)J˜ ∼ 4 for Ti, which satisfies again the Stoner condition (2.32). The inconsistency suggests strong electron correlations in transition metals, which are not described by the Hartree–Fock approximation.

36

2

Metallic Magnetism at the Ground State

2.2 Electron Correlations on Magnetism 2.2.1 Stoner Condition in the Correlated Electron System The Hartree–Fock approximation used in the last section for explaining the ferromagnetism takes into account the effects of electron–electron interactions only via the Hartree–Fock potential. The fluctuations neglected in the approximation in general tend to destroy the ordered state realized by the mean-field approximation. The effects of correlated motion of electrons leading to quantum fluctuations, which are not described by the Hartree–Fock approximation, are called electron correlations. As mentioned before in the physics of metal–insulator transition (see Sect. 1.4), electron correlations suppress electron hopping when the on-site Coulomb interaction U is large. In a simple approximation, the electron hopping energy might be renormalized as follows.     † † tij aiσ aj σ = q tij aiσ aj σ . (2.33) ij σ

0

ij σ

Here q is a phenomenological band-narrowing factor (<1), and ∼0 denotes the Hartree–Fock average. Moreover, electron correlations reduce the double occupation number n↑ n↓  on each site. We may express it as n↑ n↓  = rn↑ n↓ 0 .

(2.34)

Here r is a phenomenological reduction factor (<1). The total energy is then expressed as     † E= ε0 ni  + q tij aiσ aj σ + r U ni↑ ni↓ 0 . (2.35) i

0

ij σ

i

One can examine the stability of the ferromagnetism taking the same steps as in the Hartree–Fock approximation (see (2.1–2.5)). When there is an infinitesimal spin polarization on the Fermi level, the change in the kinetic energy E1 is given by E1 = q

(δN )2 > 0. Lρ(0)

(2.36)

On the other hand, the change in the interaction energy is given as  E2 = −rLU

δN L

2 < 0.

(2.37)

Thus the condition (2.5) for the spin polarization (E = E1 + E2 < 0) is modified as ρ(0) Ueff > 1.

(2.38)

2.2 Electron Correlations on Magnetism

37

Here Ueff denotes an effective Coulomb interaction defined by Ueff =

rU . q

(2.39)

Needless to say, it is not possible to conclude whether or not electron correlations stabilize the ferromagnetism, without knowing the band narrowing factor q and the reduction rate of the double occupancy r. In the following subsections we introduce two types of theories of electron correlations at the ground state which provide us with the correlation factors q and r.

2.2.2 Stability of Ferromagnetism in the Low Density Limit In the low density limit, electron correlations between two particles become dominant. In this subsection, we consider the correlation energy in the two electron system, and derive the effective Coulomb interaction in the low density limit [18]. Let us adopt the single-band Hubbard model (1.51) on a lattice. H = H0 + HI .

(2.40)

The noninteracting Hamiltonian H0 and the on-site Coulomb interaction HI are expressed in the momentum representation as follows.  εk nkσ , (2.41) H0 = kσ

U  † HI = ak+qσ ak† −q−σ ak  −σ akσ . 2L 

(2.42)

kk qσ

Here εk is the one electron energy eigen value for the noninteracting Hamiltonian matrix (H 0 )ij = ε0 δij + tij (1 − δij ), and nkσ is the electron occupation number operator for an electron with momentum k and spin σ . The creation (annihilation)  † † = i aiσ k|i∗ (akσ = operator in the momentum representation are defined by akσ √  i aiσ k|i), where k|i = exp(ik · R i )/ L. We now consider the two-electron state |k1 σ1 k2 σ2  in a solid such that |k1 σ1 k2 σ2  = ak†1 σ1 ak†2 σ2 |0.

(2.43)

Note that the creation operator at the r.h.s. of the equation is ordered so that the two electron states are orthogonal to each other; k1 σ1 k2 σ2 |k1 σ1 k2 σ2  = δk1 k1 δσ1 σ1 δk2 k2 δσ2 σ2 . Applying the Hamiltonian (2.40) to the two electron state, we find H |k1 σ1 k2 σ2  = (εk1 + εk2 )|k1 σ1 k2 σ2   U |k1 + q σ1 k2 − q σ2 . + δσ1 −σ2 L q

(2.44)

38

2

Metallic Magnetism at the Ground State

When σ1 = σ2 , we have HI |k1 σ1 k2 σ1  = 0. Thus |k1 σ1 k2 σ1  becomes the eigen state with no correlations. H |k1 σ1 k2 σ1  = (εk1 + εk2 )|k1 σ1 k2 σ1 .

(2.45)

The result is based on the Pauli principle that two electrons with the same spin cannot occupy the same site, so that there is no on-site Coulomb repulsion energy. When σ1 = −σ2 , the interaction term in (2.44) remains. In this case, we introduce the triplet and singlet states defined by

1 |k1 k2  = √ |k1 ↑ k2 ↓ ± |k1 ↓ k2 ↑ . 2

(2.46)

Note that {|k1 k2 } are orthogonal to each other; k1 k2 |k1 k2  = δk1 k1 δk2 k2 . Applying the Hamiltonian (2.40) onto the state {|k1 k2 } and adopting (2.44), we obtain H |k1 k2  = (εk1 + εk2 ) |k1 k2  +

U |k1 + q k2 − q. L q

(2.47)

The above equation indicates that the singlet (triplet) state |k1 k2  is scattered into {|k1 + q k2 − q}. In order to solve the energy eigen value equation, we assume that the eigen state is given as |Ψ  =



Γ (k1 k2 ) |k1 k2 .

(2.48)

k1 k2

The eigen states are classified by the total spin states. The triplet states correspond to the ferromagnetic state, while the singlet state corresponds to the nonmagnetic state. Note that |k2 k1  = ∓|k1 k2 . Thus one can verify that Γ (k2 k1 ) = −Γ (k1 k2 ) for the triplet state and Γ (k2 k1 ) = Γ (k1 k2 ) for the singlet state. Substituting the wave function (2.48) into the eigen value equation H |Ψ  = E|Ψ ,

(2.49)

we obtain (εk1 + εk2 − E)Γ (k1 k2 ) +

U Γ (k1 + q k2 − q) = 0. L q

(2.50)

 In the case of the triplet state, we can verify that q Γ (k1 + q k2 − q) = 0 using the relation Γ (k2 k1 ) = −Γ (k1 k2 ). Thus there is no Coulomb energy contribution to the triplet state. E(k1 k2 : triplet) = εk1 + εk2 .

(2.51)

2.2 Electron Correlations on Magnetism

39

Next we consider the solution for the singlet state. Starting from the singlet state |k1 k2 , we take into account the scatterings due to Coulomb interactions. Putting Γ (k1 k2 ) = 1 and defining the Coulomb energy contribution as E(k1 k2 ) = E − εk1 − εk2 , we obtain from (2.50) the following energy.    U E(k1 k2 ) = 1+ Γ (k1 + q k2 − q) . L q

(2.52)

(2.53)

The equation for Γ (k1 + q k2 − q) at the r.h.s. of the above equation is obtained from (2.50) as   

U   (εk1 +q + εk2 −q − E)Γ (k1 + q k2 − q) + 1+ = 0. Γ k1 + k k2 − k L  k =0

(2.54) Dividing the above equation by (εk1 +q + εk2 −q − E) and taking summation with respect to q, we obtain 

Γ (k1 + qk2 − q) = −

q

U G(k1 k2 ) , 1 + U G(k1 k2 )

(2.55)

where G(k1 k2 ) =

1 1 . L q εk1 +q + εk2 −q − εk1 − εk2 − E(k1 k2 )

(2.56)

Substituting (2.55) into (2.53), we obtain the interaction energy as follows. E(k1 k2 ) =

1 U . L 1 + U G(k1 k2 )

(2.57)

Alternatively, E = εk1 + εk2 +

1 U . L 1 + U G(k1 k2 )

(2.58)

The above result indicates that the Coulomb interaction U has been renormalized by the electron–electron scatterings as U/(1 + U G(k1 k2 )), since the Hartree– Fock energy in the singlet state is given by εk1 + εk2 + U/L. We then find the band narrowing factor q = 1 and the renormalization factor of Coulomb energy r = 1/(1 + U G(k1 k2 )). Note that there is no band narrowing in the low density limit because there is no electron on the surrounding sites which interrupts the motion of an electron. In conclusion, multiple scatterings of two electrons yield the

40

2

Metallic Magnetism at the Ground State

Fig. 2.3 Effective Coulomb interaction Ueff = U/(1 + U/W ) as a function of U , where W is the band width. Ubare is defined by the Hartree–Fock value Ubare = U

following effective Coulomb interaction U . Ueff =

U . 1 + U G(k1 k2 )

(2.59)

In the quantity G(k1 k2 ), we may omit E(k1 k2 ) in the denominator because E ∼ O(1/L). Moreover εk1 +q + εk2 −q − εk1 − εk2 is of order of the band width W . Thus we obtain Ueff ≈

U 1+

U W

.

(2.60)

Since ∂Ueff /∂U = 1/(1 + U/W )2 > 0, the effective Coulomb interaction Ueff monotonically increases with increasing U , but it is saturated to W as shown in Fig. 2.3. Consequently, the effective Coulomb interaction cannot exceed the band width, and the Stoner condition (2.38) is not satisfied irrespective of U for a moderate band with the band width W because ρ(0) ∼ W . ρ(0) Ueff 

1 · W = 1. W

(2.61)

The result indicates that the ferromagnetism is suppressed by electron correlations. Only for the system with large DOS at the Fermi level ρ(0), the ferromagnetism is possible. Nickel which has about 9 d-electrons per atom is regarded as a low density system according to the hole picture. The ferromagnetism of Ni is considered to be stabilized by this mechanism because of the high density of states at the Fermi level (see Fig. 2.6).

2.2.3 Gutzwiller Theory of Electron Correlations When electron number is increased, one cannot apply the low density approximation. In this subsection, we introduce the Gutzwiller variational method for electron correlations at the ground state [6–8], which is useful for any electron density.

2.2 Electron Correlations on Magnetism

41

The Hartree–Fock ground state is expressed by an independent particle state as follows. |φ =

 N↑  k

† ak↑

 N↓ 

 † ak↓

|0.

(2.62)

k

Here Nσ denotes the total electron number√for spin σ . Making use of a unitary  † ikR † = i aiσ e i / L, we obtain the wavefunction in the transformation such as akσ real space representation as follows.

|φ =

 C(N↑ , N↓

 N↓      N↑   † 1 iki ·R m(j )  † 1 iki ·R l(j ) det √ e det √ e al(i)↑ am(i)↓ |0. L L i i ) (2.63)

Here L denotes the number of lattice. k i is a momentum of the i-th electron below the Fermi level. A set of sites (l(1), l(2), . . . , l(N↑ )) ((m(1), m(2), . . . , m(N↓ ))) denote a configuration of electrons with up (down) spin on the lattice, and (R l(1) , R l(2) , . . . , R l(N↑ ) ) denote the positions of the sites. The determinants are √ defined for the matrices whose (i, j ) element is given by (1/ L) exp(ik i · R l(j ) ) √  and (1/ L) exp(ik i · R m(j ) ), respectively. Furthermore, C(N↑ ,N↓ ) means the sum over all the configurations of electrons on a lattice when electron numbers of up and down spins are given. It should be noted that the {l(i)} ({m(i)}) are ordered as l(1) < l(2) < · · · < l(N↑ ) (m(1) < m(2) < · · · < m(N↑ )) in the Fock space. In the Hartree–Fock wavefunction, doubly occupied sites appear irrespective of the Coulomb interaction strength U in various electron configurations on a lattice. Such a state with doubly occupied sites causes a loss of Coulomb interaction energy. In the correlated electron system, the probability amplitudes of doubly occupied states must be reduced to decrease the total energy. In order to describe the on-site electron correlations, Gutzwiller introduced a correlated wavefunction as follows. L  

|Ψ  = 1 − (1 − g)ni↑ ni↓ |φ.

(2.64)

i

Here ni↑ ni↓ is a projection operator that chooses the doubly occupied state on site i, and g is a variational parameter controlling the amplitudes of doubly occupied states in the Hartree–Fock wavefunction. Note that the g = 1 state corresponds to an uncorrelated state, and the g = 0 state corresponds to the atomic state in which all the doubly occupied states have been removed. Substituting (2.63) into the Gutzwiller wave function (2.64), we obtain the realspace representation as follows.

42

2

|Ψ  =



g

D

×

C(D, N↑ , N↓

 N↓ 

    N↑ 1 ik i ·R m(j )  † 1 ik i ·Rl(j ) det √ e det √ e al(i)↑ L L i ) 



D

Metallic Magnetism at the Ground State

 † am(i)↓

|0.

(2.65)

i

Here D is the number of doubly occupied sites, and C(D, N↑ , N↓ ) denotes the electron configurations on a lattice when D, N↑ , and N↓ are given. The energy for the Hubbard Hamiltonian (1.51) with ε0 = 0 is then given as follows.   † ij σ tij Ψ |aiσ aj σ |Ψ  + U Ψ | i ni↑ ni↓ |Ψ  . (2.66) E(g) = Ψ |Ψ  Each term at the r.h.s. is given as follows.    

R Ψ |Ψ  = w↑ R − R   l(1) g 2D Rl(1) D

C(D, N↑ , N↓ )

 

  Rm(1) × w↓ R − R  Rm(1)        Ψ  ni↑ ni↓ Ψ i

=

 D

Dg

2D

 C(D, N↑ , N↓ )

 · · · Rm(N↓ ) , · · · Rm(N↓ )

 

  Rl(1) w↑ R − R  Rl(1)



R × w↓ R − R   m(1) Rm(1) 

Rm(2) Rm(2)

Rl(2) Rl(2)

 · · · Rm(N↓ ) . · · · Rm(N↓ )

Rm(2) Rm(2)

· · · Rl(N↑ ) · · · Rl(N↑ )

Rl(2) Rl(2)

· · · Rl(N↑ ) · · · Rl(N↑ )



(2.67)



(2.68)

Here the function wσ (R − R  ) is defined by Nσ

1  wσ R − R  = eikn ·(R−R ) . L

(2.69)

kn

The Gutzwiller overlap function with wσ (R − R  ) in (2.67) and (2.68) is defined by       f (x1 , y1 ) f (x1 , y2 ) · · · f (x1 , yn )  f (x2 , y1 ) f (x2 , y2 ) · · · f (x2 , yn )  x1 x2 · · · xn  . (2.70) =  f (x, y) y1 y2 · · · yn ··· ··· · · ·   ··· f (xn , y1 ) f (xn , y2 ) · · · f (xn , yn ) In order to calculate the electron hopping term in the numerator of (2.66), we classify the configuration C(D, N↑ , N↓ ) into 4 parts according to the 4 electron

2.2 Electron Correlations on Magnetism

43

configurations on sites i and j ; C(D, N↑ , N↓ , i↑= j ↑= 0), C(D, N↑ , N↓ , i↑= 0, j↑= 1), C(D, N↑ , N↓ , i↑= 1, j↑= 0), and C(D, N↑ , N↓ , i↑= j↑= 1). When † aj ↑ is applied to Ψ , the configuration C(D, N↑ , the electron hopping operator ai↑ N↓ , i↑= 0, j↑= 1) remains among 4 types of configurations. The number of doubly † aj ↑ |Ψ  can changes from D according occupied states in each configuration of ai↑ to the configuration of the down spin electrons on sites i and j . When we express i↑ j ↑

the configuration as (i, j ) = i↓ j ↓ , the number of the doubly occupied states of



† ai↑ aj ↑ |Ψ  is given by D for (i, j ) = 00 10 , D − 1 for (i, j ) = 00 11 , D + 1 for



(i, j ) = 01 10 , and D for (i, j ) = 01 11 . We therefore obtain     † Ψ ai↑ aj ↑ Ψ  = g 2D 





C D, N↑ , N↓ ,(i,j )= 0 1 00

D

+



g 2D+1



+







 Rl(1) · · ·   Ri × w↑ R − R  Rj Rl(1) · · ·  

  Rm(1) Rm(2) × w↓ R − R  Rm(1) Rm(2) 



+



g 2D−1





C D, N↑ , N↓ ,(i,j )= 0 1 01

D

 C D, N↑ , N↓ ,(i,j )= 0 1 10

D





g 2D

D

Rl(N↑ ) Rl(N↑ )









C D, N↑ , N↓ ,(i,j )= 0 1 11

 · · · Rm(N↓ ) . · · · Rm(N↓ )





(2.71)

The configuration in the above expression (2.71), for example, C D, N↑ , N↓ , (i, j ) 0 1

= 0 0 means the electron configuration when D, N↑ , and N↓ are given, and there is no electron on site i, but site j is occupied by an up-spin electron. The difficulty in the Gutzwiller variational method is how to take the sums with respect to the electron configurations in each term of the energy containing the overlap functions of wσ (R − R  ). Gutzwiller replaced these overlap functions with their average values. This is called the Gutzwiller approximation. For example, in the calculation of the norm Ψ |Ψ , we make the following approximation. Ψ |Ψ  ≈

 D

g 2D

 C(D,N↑ ,N↓ )



 1   ( D C(D,N↑ ,N↓ ) )



  Rl(1) Rl(2) · · · × w↑ R − R  Rl(1) Rl(2) · · ·  

R Rm(2) · · · × w↓ R − R   m(1) Rm(1) Rm(2) · · · 





D C(D,N↑ ,N↓ )

Rl(N↑ ) Rl(N↑ )



 Rm(N↓ ) . Rm(N↓ )

(2.72)

44

2

Metallic Magnetism at the Ground State

We then obtain Ψ |Ψ  ≈

W0 (g) φ|φ. W0 (1)

(2.73)

Here φ|φ = 1. W0 (g) is a sum over all configurations of correlation weight g 2D for double occupation number. It is defined by   g 2D . (2.74) W0 (g) = D C(D, N↑ , N↓ )

In the same way, we obtain   W1↑ (g)   †     † Ψ ai↑ aj ↑ Ψ ≈ φ ai↑ aj ↑ φ , W1↑ (1)

(2.75)

W2↑ (g) φ|ni↑ ni↓ |φ. W2↑ (1)

(2.76)

Ψ |ni↑ ni↓ |Ψ  ≈ Here W1↑ (g) =





0 1

D C D, N , N ,(i,j )= ↑ ↓

+





g 2D +



0 1

D C D, N , N ,(i,j )= ↑ ↓

W2 (g) =



g

2D+1

+





g 2D−1

01



0 1

D C D, N , N ,(i,j )= ↑ ↓

10



0 1

D C D, N , N ,(i,j )= ↑ ↓

00





g 2D ,

11

(2.77) D g 2D .

(2.78)

D C(D, N↑ , N↓ )

The weighting functions W0 (g), W1 (g), and W2 (g) are expressed by the following type of the hypergeometric functions.

F α − N↑ , β − N↓ , L − N + γ ; g 2 =

∞  (L − N + γ − 1)!(α − N↑ + D − 1)!(β − N↓ + D − 1)! g 2D . (2.79) D!(L − N + γ + D − 1)!(α − N↑ − 1)!(β − N↓ − 1)!

D=0

Here N is the total number of electrons. α, β, and γ are integers of order of 1. In the evaluation of the sums over D in these functions, we can adopt the maximum term approximation because D is a macroscopic variable. The weights for each D term is proportional to (α − N↑ + D − 1)!(β − N↓ + D − 1)! g 2D . D!(L − N + γ + D − 1)!

(2.80)

2.2 Electron Correlations on Magnetism

45

The representative value of D that maximizes the above factor is given by g2 =

D(L − N↑ + D) . (N↑ − D)(N↓ − D)

(2.81)

By making use of the representative value D, we can express the total energy as follows. N  σ   E(g) = qσ εk + U D. (2.82) σ

k

Here the band narrowing factor qσ is given as 

2 √ (Nσ − D)(L − N + D) + D(N−σ − D) . qσ = Nσ (L − Nσ )

(2.83)

In the nonmagnetic state at half-filling (N = L), the energy per site is simplified as follows.

Here

ε(g) = −q|εb | + U d.

(2.84)

 1 − d d, q = 16 2

(2.85)



g=

d 1 2

−d

,

(2.86)

and d = D/L denotes the double occupation number per site. εb is the band energy per site, and is given by the noninteracting density of states per atom and per spin ρ(ε) = L−1 k δ(ε − εk ) as follows.  0 εb = 2 ερ(ε) dε. (2.87) −∞

Minimizing the energy (2.84) with respect to g (i.e., d), we obtain U2 , Uc2   U 1 , 1− d= 4 Uc q =1−

(2.88) (2.89)

and U Uc g= . U 1+ Uc 1−

(2.90)

46

2

Metallic Magnetism at the Ground State

Here Uc = 8|εb | is a critical Coulomb interaction at which q and d vanish. The ground-state energy is given by   1 U 2 ε(g) = − Uc 1 − . (2.91) 8 Uc When U > Uc , there is another solution: g = 0 or d = 0 which yields the minimum energy ε(g) = 0. Note that electrons are completely localized at U = Uc ; q = d = 0. This implies that the metal-insulator transition occurs at U = Uc . Equation (2.89) means that the rate of double occupation r = d/(n↑ 0 n↓ 0 ) is given by U r =1− . (2.92) Uc By making use of (2.88) and (2.92), we obtain the effective Coulomb interaction for the ferromagnetic instability which is defined by (2.39), i.e., Ueff = rU/q. U Ueff = . (2.93) U 1+ Uc Since Uc = 2W for the rectangular DOS with the band width W , the expression above is essentially the same as the effective Coulomb interaction (2.60) in the low density approximation. We therefore again find that the electron correlations suppress the ferromagnetism in the case of usual electron density as well. In the above analysis, we omitted the polarization via qσ and r. More detailed calculations of spin susceptibility yield [19] m∗ 2ρ(0) me . (2.94) χ= 1 − ρ(0)U˜ eff Here m∗ /me = q −1 is the effective mass of electrons and U˜ eff is defined by   U U 1+ U 2Uc . (2.95) U˜ eff =  2 ≈ U U 1 + 1+ Uc Uc Note that the susceptibility diverges at the metal-insulator point U = Uc via effective mass due to the formation of the atomic state with spin s = 1/2 in the Gutzwiller approximation. The effective Coulomb interaction U˜ eff is associated with the formation of the ferromagnetism via the denominator in (2.94), and is essentially the same as Ueff given by (2.93). The theory mentioned above is based on the single band model. More realistic calculations on the stability of the ferromagnetism have been made on the basis of the five d band model and the local ansatz variational method; the latter takes into account the on-site Hund’s rule correlations as well as the density correlations [20]. The degeneracy of the bands in general increases the channel of electron hopping, so that the correlation effects tend to be reduced. The effective Coulomb interac-

2.3 Density Functional Approach

47

tions for the stability of the ferromagnetism are typically reduced by 30–40 % as compared with the bare values in the case of 3d transition metals. One of the important results of the theory is that the energy difference between the ferromagnetic state and the nonmagnetic state is much reduced by electron correlations. In case of Fe, for example, the difference is 0.56 eV per atom in the Hartree–Fock theory, reduced to 0.22 eV when the density correlations are taken into account, and reduced even further to 0.15 eV when both density and spin correlations are taken into account [21].

2.3 Density Functional Approach 2.3.1 Slater’s Band Theory Towards a realistic description of magnetic properties in metals and alloys, it is desirable to calculate one electron state in solids quantitatively. A reasonable one electron state may be the Hartree–Fock wave function. As we have mentioned in Sect. 1.3, the wavefunctions are determined by the following Hartree–Fock selfconsistent equation at the ground state.    n(r  ) 1 ϕiσ (r) − ∇ 2 + vN (r) + dr  2 |r − r  | occ   ϕj∗σ (r  )ϕiσ (r  ) ϕj σ (r) = εiσ ϕiσ (r). (2.96) − dr  |r − r  | j

 Here εiσ is the Hartree–Fock one-electron energy eigen value. occ denotes a sumoccj ∗ mation over occupied Hartree–Fock orbital {j }, and n(r) = j σ ϕj σ (r)ϕj σ (r) denotes a charge density in the Hartree–Fock approximation. The third term at the l.h.s. expresses the electrostatic potential due to electrons. The last term called the exchange potential originates in the Coulomb interaction and the anti-symmetric property of the Slater determinant. The Hartree–Fock total energy is given by HHF  =

occ 

εiσ −

iσ

1  (Vijj i − Vij ij δσ σ  )niσ nj σ  . 2 

(2.97)

ij σ σ

Here Vijj i (Vij ij ) is the Coulomb (exchange) energy integral. Because of the exchange potential, the self-consistent equation is non-linear, so that it is not easy to solve the equation in solids. In order to simplify the exchange term, Slater rewrote it as occ   ϕ ∗ (r  )ϕiσ (r  )  jσ ϕj σ (r) dr − |r − r  | j

=−

 occ   j

dr



∗ (r)ϕ (r) ϕj∗σ (r  )ϕiσ (r  )ϕiσ jσ ∗ (r)ϕ (r)|r − r  | ϕiσ iσ

 ϕiσ (r).

(2.98)

48

2

Metallic Magnetism at the Ground State

The term [∼] at the r.h.s. of the above equation is regarded as an effective potential. Slater [4] took an averageof the effective potential with respect to the orbital i with ∗ (r)ϕ (r)/( occ ϕ ∗ (r)ϕ (r)), so that the exchange term is written as a weight ϕiσ iσ lσ l lσ −

occ   j

dr 

ϕj∗σ (r  )ϕiσ (r  ) |r − r  |



n˜ ex σ r, r  =

 ϕj σ (r) ≈ −

dr 

n˜ ex σ (r, r  ) ϕiσ (r), |r − r  |

(2.99)

occ occ i

j

∗ (r  )ϕ (r  )ϕ ∗ (r)ϕ (r) ϕiσ jσ iσ jσ occ ∗ . ϕ (r)ϕ (r) jσ j jσ

(2.100)

The Hartree–Fock equation is then reduced to a linear equation as follows.    1 n(r  ) (HFS) − ∇ 2 + vN (r) + dr  + v (r) ϕiσ (r) = εiσ ϕiσ (r). (2.101) ex σ 2 |r − r  | (HFS)

Here vex σ (r) is the Hartree–Fock–Slater exchange potential defined by  n˜ ex σ (r, r  ) (HFS) vex . (r) = − dr  σ |r − r  |

(2.102)

Slater simplified further the potential by using the free electron wave functions as follows [4].  vex σ (r) = −3

3 4π

1/3 nσ (r)1/3 ,

(2.103)

where nσ (r) is the electron density for spin σ . For more practical use, Slater proposed the following potential with an adjustable parameter α.  vXα σ (r) = −3α

3 4π

1/3 nσ (r)1/3 .

(2.104)

This is known as the Xα potential. Numerical calculations suggest that α = 1/2–2/3 is suitable for explaining various experiments.

2.3.2 Density Functional Theory As discussed before, the orthodox approach to the electronic structure and magnetism in solids may be to first apply the Hartree–Fock theory leading to the best one electron wavefunction at the ground state, and then taking into account the correlation corrections. This scheme however seems to be inconvenient for quantitative calculations, firstly because the Hartree–Fock equation is too complicated due to nonlinear exchange potentials, secondly because the Hartree–Fock approximation

2.3 Density Functional Approach

49

is not necessarily a good starting point to describe the magnetism as mentioned before in Sect. 2.2. We have discussed a simplified Hartree–Fock–Slater potential for practical use of the Hartree–Fock approach in the last subsection. Kohn, Sham, and Hohenberg proposed the density functional theory (DFT) which justifies Slater’s idea for the exchange potential [22–25]. The theory allows us to include electron correlations in solids in a simple manner. The DFT is based on the Hohenberg–Kohn theorem [22, 24]. It states that the ground state energy E0 is given by the minimization of a density functional E[n]: E0  E[n],  E[n] = F [n] + vext (r)n(r) dr,     F [n] = Ψnmin (Tˆ + Vˆee )Ψnmin ,  Zl . vext (r) = − |r − R l |

(2.105) (2.106) (2.107) (2.108)

l

Here n(r) is an electron density. Tˆ (Vˆee ) is the kinetic energy (electron–electron interactions). Ψnmin is the wave function that minimizes the energy under a given density n(r), and vext (r) is the electron-nuclear interaction as an external field. Note that the energy of the electron system F [n] is a universal functional which does not depend on vext (r). In order to verify the inequality in (2.105), let us express the ground state energy as        E0 = Min{Ψ } Ψ (Tˆ + Vˆee )Ψ + vext (r)n(r) dr .

(2.109)

Here Min{Ψ } means to choose from possible Fermion wave functions {Ψ }, the ground-state wavefunction that minimizes the total energy. When the N electron Hilbert space is partitioned by the wave functions with the same density n, the above expression is written as follows (see Fig. 2.4).  E0 = Min{n}

    Min{Ψ |n} Ψ (Tˆ + Vˆee )Ψ +



 vext (r)n(r) dr .

(2.110)

The notation Min{Ψ |n} means to minimize the following term by choosing a wavefunction among those which yield the same density n. The first term at the r.h.s. of (2.110) is given by F [n], which is defined by (2.107), thus the above equation leads to the inequality (2.105). Kohn and Sham [23] assumed that the many-body electron density is reproduced by the density of noninteracting electrons with an effective potential v(r), and wrote

50

2

Metallic Magnetism at the Ground State

Fig. 2.4 The N electron Hilbert space is partitioned by the wave functions with the same density n

the energy F [n] as follows. F [n] = Ts [n] +

1 2



n(r)n(r  ) dr dr  + EXC [n]. |r − r  |

(2.111)

The first term Ts [n] at the r.h.s. is a kinetic energy for a noninteracting system     occ   occ   1 2   Ts [n] = εi − v(r)n(r) dr. (2.112) ψ i  − ∇ ψ i = 2 i

i

occ

Here n(r) = i |ψi (r)|2 is a density of noninteracting electrons, and ψi (r) is a one-electron eigen function for the eigen value εi .   1 − ∇ 2 + v(r) ψi (r) = εi ψi (r). (2.113) 2 It should be noted that the kinetic energy Ts [n] is not identical with T [n] = Ψ |Tˆ |Ψ ; Ts [n] − T [n] has been included in the exchange correlation energy EXC [n]. The electron number is given by  n(r) dr = N. (2.114) When the universal functional EXC [n] is assumed to be known, one can determine the effective potential v(r) from the variational principle.

δ E[n] − μN = 0, (2.115) where μ is the chemical potential. Substituting (2.106) and (2.111) into the above equation and taking the variation, we obtain  n(r  ) dr v(r) = vext (r) + + vxc (r) − μ, (2.116) |r − r  | and the exchange-correlation potential vxc (r) is defined by vxc (r) =

δEXC . δn(r)

(2.117)

2.3 Density Functional Approach

51

Equations (2.113) and (2.116) are referred as the Kohn–Sham equations. Using (2.116), the total energy is given by E[n] =

occ  i

εi −

1 2



n(r)n(r  ) dr dr  + EXC [n] − |r − r  |

 vxc (r)n(r) dr + μN. (2.118)

The Kohn–Sham theory provides us with a method to calculate the density and the total energy of correlated electron system by solving one-electron Schrödinger equation once we know a universal exchange-correlation energy EXC [n]. The Hohenberg–Kohn–Sham method can be extended to the spin polarized case. In this case, we introduce a magnetic field H (r) coupled to the spin density m(r). The energy becomes a functional of the charge and spin densities: E[n, m], and  E[n, m] = F [n, m] +

E0  E[n, m],

(2.119)



vext (r)n(r) − m(r) · H (r) dr.

(2.120)

The universal functional F [n, m] in the Kohn–Sham method is expressed as  1 n(r)n(r  ) dr dr  + EXC [n, m]. F [n, m] = Ts [n, m] + (2.121) 2 |r − r  | Introducing the wave function ψk (r) = (ψk↑ (r), ψk↓ (r)) and the spin dependent potential Δ(r) = (Δx (r), Δy (r), Δz (r)), the Kohn–Sham one-electron eigen value equation is given by   1 − ∇ 2 + v(r) − Δ(r) · σ ψk (r) = εk ψk (r). (2.122) 2  2 The and spin densities are obtained by n(r) = occ kσ |ψkσ (r)| and m(r) = occcharge ∗ kσ σ  ψkσ (r)(σ )σ σ  ψkσ  (r), respectively. The potentials obtained by the variational principles are given by (2.116) and Δ(r) = Δxc (r) + H (r).

(2.123)

The exchange correlation potentials vxc (r) and Δxc (r) are defined by vxc (r) =

δEXC [n, m] , δn(r)

Δxc (r) =

δEXC [n, m] . δm(r)

(2.124)

The ground-state energy in the Kohn–Sham scheme is then given by E[n, m] =

occ  k

−



εk −

1 2



n(r)n(r  ) dr dr  + EXC [n, m] |r − r  |

vxc (r)n(r) − Δxc (r) · m(r) dr + μN.

(2.125)

52

2

Metallic Magnetism at the Ground State

Actual expressions of the exchange correlation potentials have been obtained by many investigators. The simplest way is to assume that the exchange correlation energy is a function of the electron densities and adopt an approximate form obtained from the homogeneous electron gas system. 

(2.126) EXC [n] = n(r)εxc n(r) . Here εxc (n(r)) is the exchange correlation energy per electron for a homogeneous electron gas. This is called the Local Density Approximation (LDA). The LDA is referred as the Local Spin Density Approximation (LSDA) when spin polarization is taken into account. The exchange-correlation energy εxc (n) consists of the exchange energy part εx (n) and the correlation part εc (n); εxc (n) = εx (n) + εc (n). The exchange energy in the electron gas is given by   3 3n 1/3 εx (n) = − . 4 π

(2.127)

This yields the exchange potential in the paramagnetic state.  vxP (n) = −

3n π

1/3 =−

0.611 [a.u.]. rs

(2.128)

where rs denotes a radius for the sphere with the volume per electron, i.e., rs = (4πn/3)−1/3 . When the correlation term is neglected, the potential reduces to the Slater exchange potential (2.103) with the difference by a factor α = 2/3. Barth and Hedin [26] proposed the exchange correlation potential with use of the result of the random phase approximation for electron gas. Assuming the ferromagnetic spin polarization, the potential vσxc (r) = vxc (r) − Δz (r)σ is given as follows. vσxc (r) = vσx (r) + vσc (r).

(2.129)

The exchange part is given by vσx (r) = vxP

 

F nσ P . + vx − vx f n

(2.130)

Here vxF = −0.770/rs is the exchange potential for the complete polarization state (n↑ = n) and the function f (x) is defined by f (x) =

 4/3  1 x + (1 − x)4/3 − 2−1/3 . −1/3 1−2

(2.131)

Note that f (1/2) = 0 in the paramagnetic state and f (0) = 1 in the complete ferromagnetic state.

2.3 Density Functional Approach

53

Fig. 2.5 Exchange and correlation energies (εx = εx − εxP and εc = εc − εcP ) per electron as a function of spin polarization ζ = (n↑ − n↓ )/n (case of rs = 2) [27]. VWN: εc by Vosko et al., GL: εc by Gunnarsson et al.

The correlation potential vσc (r) is expressed as vσc (r) = vcP

 

F nσ P . + vc − vc f n

(2.132)

The correlation potentials in the paramagnetic and ferromagnetic states are defined by     30 75 0.0504 0.0254 P F vc = − , vc = − . (2.133) ln 1 + ln 1 + 2 rs 2 rs Figure 2.5 shows the energy gain of correlation energy per electron as a function of spin polarization ζ = (n↑ − n↓ )/n. As expected from our discussions in the previous sections, the spin polarization is caused by the exchange energy gain, but the correlation energy acts to suppress polarization. This is the same physics as discussed in the last subsection. It should be noted that in the local orbital representation the on-site Coulomb interaction was very large (∼20 eV). In the LSDA based on the homogeneous electron gas the contribution from the Coulomb integral on the same orbital is negligible because the wave functions for electron gas are extended over the crystal.

2.3.3 Tight-Binding Linear Muffin-Tin Orbitals In the LDA scheme, calculations of the ground-state properties in solids are reduced to an eigen-value problem for one electron Hamiltonian H = −∇ 2 /2+v(r) as given

54

2

Metallic Magnetism at the Ground State

by (2.113): H ψi (r) = εi ψi (r).

(2.134)

Among various methods to solve the eigen value equation in solids, the TightBinding Linear Muffin-Tin Orbital (TB-LMTO) method [28–30] is compatible with the tight-binding model which has been used in the previous sections, and allows us the first-principles calculations of the Hamiltonian matrix elements in real space. In this subsection, we describe the TB-LMTO method for electronic structure calculations. The basic idea of the TB-LMTO method is found in the one electron eigen value problem of a hydrogen molecule. The eigen states are approximately given by a linear combination of atomic orbitals ϕ(r) at the origin and ϕ(r − R) at another position of hydrogen atom R. ϕB (r) = ϕ(r) + ϕ(r − R),

(2.135)

ϕA (r) = ϕ(r) − ϕ(r − R).

(2.136)

Here ϕB (r) (ϕA (r)) is the bonding (antibonding) orbital of the hydrogen molecule leading to the eigen value EB (EA ). Solving the above equations with respect to ϕ(r) and ϕ(r − R), we can express the atomic orbitals as

1 ϕB (r) + ϕA (r) , 2

1 ϕ(r − R) = ϕB (r) − ϕA (r) . 2

ϕ(r) =

(2.137) (2.138)

We introduce here an energy-dependent wave function for the hydrogen molecule ϕE (r) such that H ϕE (r) = EϕE (r) where H is the one-electron Hamiltonian of the molecule, and rewrite the above equations as ϕ(r)  ϕE0 (r) + c1 ϕ˙ E0 (r),

(2.139)

ϕ(r − R)  c2 ϕ˙ E0 (r).

(2.140)

Here E0 is an energy such that EB ≤ E0 ≤ EA , and ϕ˙ E0 (r) = ∂ϕE0 (r)/∂E. The above expressions of atomic orbitals suggest that the atomic orbitals as the basis functions for solids can be expressed by a linear combination of the energy dependent wave functions ϕE0 (r) and their derivatives ϕ˙E0 (r). Let us assume that the crystalline potential consists of the spherical potentials centered at each nucleus and the flat potential in the interstitial region. The former potentials are called the muffin-tin (MT) potentials. We then define the following atomic orbital on site i and orbital L = (l, m) as a basis function of solid.  α i χiL (r − R i ) = ϕiL (r − R i ) + χiL (r − R i ) + ϕ˙ jαL (r − R j )hαjL iL , (2.141) j L

2.3 Density Functional Approach

55

α α ϕ˙ iL (r − R i ) = ϕ˙ iL (r − R i ) + ϕiL (r − R i )oiL .

(2.142)

α (r − R ) are called the LMTO. Here the spin part has been The wave functions χiL i omitted for simplicity. The atomic wave function ϕiL (r − R i ) is defined to satisfy the Schrödinger equation with an energy EνiL inside the MT sphere i with radius siR ; (H − EνiL )ϕiL (r − R i ) = 0, and vanishes outside the sphere i. Because of the spherical potential inside the MT sphere, it is written as ϕiL (r) = φiL (EνiL , r)YL (ˆr ) using the cubic harmonics YL (ˆr ). φiL (E, r) is determined by the radial Schrödinger equation with energy E. The energy EνiL is usually chosen to be the center of the gravity of energy eigen values below the Fermi level for each orbital. Note that ϕiL ’s are normalized in the MT sphere as ϕiL |ϕj L  = δij δLL . i (r − R ) is an atomic wave The second term at the r.h.s. of (2.141), i.e., χiL i function for the interstitial region that satisfies the Schrödinger equation in the ini = 0 with k = 0, so that the function is expressed by terstitial region; (∇ 2 + k 2 )χiL the irregular solution of the Laplace equation as follows.

 i (r − R i ) = KiL (r − R i ) = χiL

|r − R i | siR

−l−1

YL (r − R i ).

(2.143)

i (r − R ) inside the MT spheres is defined to be zero. The wave function χiL i Inside each MT sphere (j = i), the basis wave function is described by the α (r − R ) is defined by (2.142). third term at the r.h.s. of (2.141) where ϕ˙ iL i ϕ˙ iL (r −R i ) is the energy derivative of ϕiL (r −R i ) at Eνil ; ϕ˙iL (r −R i ) = [∂ϕiL (r − α (r − R ) deR i )/∂E]E=EνiL , and ϕiL |ϕ˙j L  = 0. Note that the wave function ϕ˙ iL i α scribing the tails of the LMTO contains an arbitrary parameters oiL which allows us to control the localization of the basis function. The coefficients hαjL iL in the tail part of (2.141) are determined so that the wave i (r − R ) in the interstitial region is smoothly connected to those inside function χiL i the MT spheres. This can be performed by using an envelope function ∞ (r − R i ) = KiL (r − R i ) − KiL



JjαL (r − R j )SjαL iL ,

(2.144)

j L i (r − R ) inside all the which smoothly connects the interstitial wave function χiL i α MT spheres. JiL is a linear combination of the regular solution JiL and the irregular solution KiL of the Laplace equation being defined by α JiL (r − R i ) = JiL (r − R i ) − KiL (r − R i )αiL .

(2.145)

α The matrix SiLj L in (2.144) is the so-called screened structure constant which is expressed by the canonical structure constant SiLj L as

  α −1 SiLj . L = S(1 − αS) iLj L

(2.146)

56

2

Metallic Magnetism at the Ground State

Here SiLj L is given by    √ |r − R j | −l−l −1 ∗ Sj L iL = 4π gl  m lm Yl+l  m −m (R i − R j ), sw

√ gl  m lm = (−)l+1 4π

2(2l  − 1)!! Clm l  m l  m , (2l − 1)!!(2l  − 1)!!

(2.147)

Clm l  m l  m being the Gaunt coefficient, and sw is an arbitrary length such as the average Wigner–Seitz radius of the lattice. The parameters αiL screen the structure constant SjαL iL as seen from (2.146). With the use of the matching condition on each muffin-tin sphere, we obtain the α via ϕ˙ α as relation between αiL and oiL iL αiL =

(siR /sw )2l+1 D{ϕ˙ α } − l . 2(2l + 1) D{ϕ˙ α } + l + 1

(2.148)

α is expressed by α as Alternatively oiL iL α oiL =−

˙ − wiL {K, ϕ}α ˙ iL wiL {J, ϕ} . wiL {J, ϕ} ˙ − wiL {K, ϕ}αiL

(2.149)

Here wiL {a, b} is the Wronskian defined by   wiL {a, b} = siR aiL (siR )biL (siR ) D{biL } − D{aiL } , D{a} = siR a  (siR )/a(siR ).

(2.150)

α Finally the tail function hαiLj L is obtained from the global matching between χiL ∞ and KiL on the muffin-tin spheres.

hαiLj L

wiL {K, ϕ} δij δLL + =− wiL {K, ϕ α }



 ! α ! 2 2 α α wiL J , ϕ SiLj L wj L J , ϕ . sw sw (2.151)

α = 0 corresponds to the choice of α = γ According to (2.148), the choice oiL iL iL such that

γiL =

˙ −l (siR /sw )2l+1 D{ϕ} . 2(2l + 1) D{ϕ} ˙ +l+1

(2.152)

The choice of αiL = γiL is called the nearly orthogonal representation (or γ -representation) because in this case the MTO’s become orthogonal to each other γ up to the second order in hiLj L . Andersen et al. [28] showed that the followα (r − R )}; α = 0.34850, ing choice of αiL yields well localized basis set {χiL i is αip = 0.05303, αid = 0.010714, and αi l(3) = 0.0. The MTO’s with such a set of {αiL } are called the TB-LMTO.

2.3 Density Functional Approach

57

In the atomic sphere approximation (ASA) in which the Wigner–Seitz cells are replaced by the atomic spheres with the same volume, the interstitial region disappear, so that the basis function reduces to  α χiL (r − R i ) = ϕiL (r − R i ) + ϕ˙jαL (r − R j )hαjL iL . (2.153) j L

In the γ representation, we have γ

χiL (r − R i ) = ϕiL (r − R i ) +

 j L

γ

ϕ˙j L (r − R j )hj L iL .

(2.154)

α are parameterized by the position potenThe coefficients hαiLj L and oiL α α α |H |χ α , and the parameter Δα = tial parameter ciL defined by ciL = χiL iL iL 1/2 α −(2/sw ) wiL {J , ϕ} as follows.

1/2 α α 1/2 α − EνiL δij δLL + ΔαiL SiLj L Δj L , (2.155) hαiLj L = ciL α =− oiL

ΔαiL

γiL − αiL . α −E + (γiL − αiL )(ciL νiL )

(2.156)

In the next step, we construct the Hamiltonian matrix. Using the basis set of γ representation (2.154), we obtain the matrix elements of the Hamiltonian HiLj L and overlap integral OiLj L as follows.  γ  γ  γ  γ γ γ γ hiLkL EνkL pkL hkL j L , HiLj L = χiL H χj L = hiLj L + EνiL δij δLL + kL

γ

OiLj L

 γ γ  γ  γ γ = χiL χj L = δij δLL + hiLkL pkL hkL j L . 

(2.157) (2.158)

kL

γ

2  ≡ ϕ˙ |ϕ˙ . In the matrix form, they are written as Here piL = ϕ˙iL iL iL

H γ = hγ + E ν + hγ E ν pγ hγ ,

(2.159)

O γ = I + hγ p γ hγ .

(2.160)

Similarly, the matrix elements of the Hamiltonian HiLj L and overlap integral OiLj L in the α representation are given as follows.



H α = I + hα oα hα + I + hα oα E ν I + oα hα + hα E ν p α hα , (2.161)



O α = I + hα oα I + oα hα + hα pα hα . (2.162) Note that we can switch the basis set {χ α } into the set {χ γ } via the following relations,  

−1  γ χjαL I + oα hα , (2.163) χiL = j L iL j L

58

2

Metallic Magnetism at the Ground State

and

−1 hγ = hα I + oα hα = hα − hα oα hα + · · · .

(2.164)

Using the above expressions, the Hamiltonian in the completely orthogonal representation is given by H = O γ −1/2 H γ O γ −1/2 = O α −1/2 H α O α −1/2 .

(2.165)

The Hamiltonian matrix element in this representation is given as follows to the second order in E − EνiL .

γ HiLj L = EνiL δij δLL + hiLj L = EνiL δij δLL + hαiLj L − hα oα hα iLj L . (2.166) Thus, the Hamiltonian in the nearly orthogonal basis set has the same form as Slater’s two-center tight-binding Hamiltonian [31]. HiLj L = εiL δij δLL + tiLj L (1 − δij δLL ). γ

(2.167)

γ

Here εiL = EνiL + hiLiL and tiLj L = hiLj L = hαiLj L − (hα oα hα )iLj L . This is the first-principles TB-LMTO Hamiltonian. In the actual band calculations, we solve the eigen value equation in the α representation.   α α HiLj OiLj (2.168) L uj L (k) = εn (k) L uj L (k). j L

j L

Here εn (k) is the energy eigen value for the momentum k and the quantum number n. uiL (k) is the eigen vector in the α representation. The self-consistent calculations of potential are also possible by recalculating the charge (as well as spin) density within the atomic sphere after we obtain eigen values and the wave functions in the atomic sphere.

2.3.4 Ferromagnetism in Transition Metals Using the TB-LMTO and ASA, one can obtain the effective exchange energy parameter called the Stoner parameter. Let us assume that one-electron energy eigen values for spin σ are given by εnσ (k). The exchange splitting for the electron with quantum number (n, k) is given by εnσ (k) = εn↓ (k) − εn↑ (k). The average exchange splitting near the Fermi level is therefore given by 1 ε = Nρ(εF )Δ

εF ≤εnσ (k)≤εF + nk

εnσ (k).

(2.169)

2.3 Density Functional Approach

59

Here N is the number of atoms. ρ(εF ) is the total DOS per atom per spin at the Fermi level, and Δ is an infinitesimal width of energy at the Fermi level. The exchange splitting is caused by the exchange correlation potential vσxc (r) = δE  XC /δnσ (r). In the LDA, the exchange correlation potential has the form EXC = sR nεxc (n, m) dr. Defining G(n, m) by G(n, m) = nεxc (n, m),

(2.170)

the exchange correlation potential is given as follows for a small polarization m(r). vσxc (r) = Gn (n, 0) + Gm (n, 0)mσ.

(2.171)

Here Gn (n, m) = ∂G/∂n and Gm (n, m) = ∂ 2 G/∂m2 . Then we obtain the splitting εnσ (k) as     εnσ (k) = −2 ψn (k)Gm (n, 0)m(r)ψn (k) . (2.172) Substituting εnσ (k) from (2.172) into (2.169) and using the ASA, we obtain ε =  Iil = − 0

1  ρil (εF ) , Iil mi N ρ(εF )

(2.173)

il

sR

2Gm (n, 0)

mi (r) φil (εF , r)2 r 2 dr. mi

(2.174)

Here mi (mi (r)) is the magnetic moment (the spin density) on site i. ρil (εF ) is the partial DOS the orbital l per spin in the nonmagnetic state, and is defined by  for ρil (εF ) = m nk χiL |ψn (k)δ(ε − εn (k))ψn (k)|χiL . The Stoner parameter Ii on site i may be defined by 1  Ii mi . N

ε =

(2.175)

i

Comparing (2.175) with (2.173), we find Ii =



Iil

l

ρil (εF ) . ρ(εF )

(2.176)

This is the exchange energy parameter derived from the DFT-LDA theory, i.e., the Stoner parameter. The spin density mi (r) on site i in (2.174) is given by mi (r) = ni↑ (r) − ni↓ (r), and each niσ (r) is given by  εF  1 niσ (r) = dε φilσ (ε, r)2 ρilσ (ε). (2.177) 4π l

For small polarization, we have a polarized wave function φilσ (ε, r) = φil (ε + εσ/2, r) and the polarized DOS ρilσ (ε) = ρil (ε + εσ/2) according to the

60

2

Table 2.1 Stoner parameters and Stoner criterions in transition metals [32]

Metallic Magnetism at the Ground State

Metal

V

Fe

Co

Ni

Pd

Pt

I (eV)

0.80

0.92

0.99

1.01

0.70

0.63

ρ(εF )I

0.9

1.6

1.7

2.1

0.8

0.5

Schrödinger equation with constant exchange splitting ε. We therefore obtain the magnetization mi (r) = ni↑ (r) − ni↓ (r) from (2.177) as follows. ρil (εF ) 1  mi (r) . = φil (εF , r)2 mi 4π ρi (εF )

(2.178)

l

Here ρi (ε) is the local DOS per spin on site i. Substituting (2.178) into (2.174), we obtain  siR 1

ρil  (εF )  2 2 2 Iil = −2Gm (n, 0) φil  (εF , r) φil (εF , r) r dr . 4π ρi (εF ) 0 

(2.179)

l

In transition metals, the d component is dominant in the total DOS at the Fermi level, so that we obtain  sR 1 Gm (n, 0)φil (εF , r)4 r 2 dr. (2.180) I =− 2π 0 The Stoner condition for the ferromagnetism in the LDA-DFT is therefore given by ρ(εF )I > 1.

(2.181)

Here ρ(εF ) is the total DOS per atom per spin. Gunnarsson calculated the Stoner parameters of transition metals [32]. The results are presented in Table 2.1. Calculated Stoner parameters for Fe, Co, and Ni are 0.92, 0.99, and 1.01 eV, respectively. These values are close to the exchange energy parameters in atoms obtained by the Hatree–Fock calculations: 0.88, 0.94, and 0.99 eV, but they are much smaller than the effective exchange energy parameters in the Hatree–Fock approximation J˜ = U0 /5 + 4J /5 (see (2.28)): 5.7, 6.0, and 6.4 eV, respectively, according to the Hatree–Fock atomic calculations. The discrepancy is due to a large intra-atomic Coulomb interaction energy between the same orbitals (U0 ∼ 20 eV). The Hartree–Fock approximation is not a good starting point for such a condition. The electron correlations reduce this to Ueff ∼ W as discussed in Sect. 2.2, so that one can expect that the effective J˜ becomes comparable to those obtained from the DFT-LDA. It should be noted that the intra-orbital Coulomb interactions are negligible in the free electron gas, so that the large U0 term as found in the atomic calculations does not appear in the DFT-LDA scheme. The densities of states (DOS) in the paramagnetic state of 3d transition metals are presented in Fig. 2.6. The bcc DOS is characterized by a two-peak structure and a deep valley between the main peaks. The Fermi level of V is near the top of

2.3 Density Functional Approach

61

Fig. 2.6 Densities of states of bcc Fe (solid curve) and fcc Ni (dotted curve). The Fermi levels of bcc V, bcc Cr, fcc Mn, and fcc Co are shown by the arrows

Table 2.2 Theory vs. experiment of the ground-state magnetizations M for Fe, Co, and Ni [34]. Theoretical results are the spin contribution to the magnetizations calculated at the experimental lattice constant. The values in the parentheses in the experimental data denote the spin contribution M (μB )

Fe

Co

Ni

LDA

2.15

1.51

0.59

Expt.

2.22 (2.12)

1.74 (1.71)

0.62 (0.60)

the second peak. The Fermi level of Cr is near the bottom of the valley. The Fermi level of the bcc Fe is located on the main peak, so that bcc Fe is favorable to the ferromagnetism. The fcc DOS is characterized by a high DOS in the main peak, which is caused by the t2g d bands. The Fermi level of the fcc Ni is on the peak, and thus it is favorable for the strong ferromagnetism. The Stoner parameters as well as the DOS at the Fermi level obtained by the LSDA explain the ferromagnetism in bcc Fe, Co, and Ni, and the paramagnetism in V, Pd, and Pt as shown in Table 2.1. Janak [33] performed the spin susceptibility calculations for 32 metallic elements from Li to In on the basis of the DFT-LDA, and demonstrated that the other elements do not satisfy the Stoner criterion as they should. Moruzzi et al. [34] calculated the ground-state magnetization of transition-metal ferromagnets. As shown in Table 2.2, the LDA quantitatively explains the groundstate magnetization in Fe, Co, and Ni. The LDA, however, does not explain the crystal structure of Fe correctly; the nonmagnetic fcc structure is stabilized for Fe in the LDA. Perdew et al. [35–37] developed a method which self-consistently takes into

62

2

Metallic Magnetism at the Ground State

account the gradient terms of the spin and charge densities being neglected in the LDA. The method called the generalized gradient approximation (GGA) improves the structural properties of solids. The GGA yields the ground-state magnetizations 2.17 μB for bcc Fe, 1.66 μB for fcc Co, and 0.66 μB for Ni at calculated lattice constants [38]. The ground-state magnetization in Ni is somewhat overestimated. In general the GGA enhances the magnetization as compared with the LDA.

Chapter 3

Metallic Magnetism at Finite Temperatures

When temperature is elevated, various excited states of electrons appear in the ensemble average of magnetic moments. In this chapter, we present the theories of metallic ferromagnetism at finite temperatures. We first deal with the finitetemperature ferromagnetism by means of the Hartree–Fock approximation called the Stoner theory, and point out that the theory overestimates the Curie temperature due to the lack of spin fluctuations. The single-site spin fluctuation theory (SSF) describes thermal spin fluctuations, and significantly improves the Stoner theory. We introduce in Sect. 3.2 the functional integral method (FIM) which describes spin fluctuations at finite temperatures with use of the auxiliary exchange fields, and derive the SSF in Sect. 3.3 on the basis of the FIM. The SSF yields the Curie–Weiss susceptibility, and explains various properties of metallic magnetism at finite temperatures qualitatively. But dynamical spin and charge fluctuations are not taken into account in the theory because it is based on the high temperature approximation called the static approximation. We present the dynamical CPA in Sects. 3.4 and 3.5, which completely takes into account the dynamical spin as well as charge fluctuations at finite temperatures within the single-site approximation. We discuss the dynamical effects on the metallic magnetism. The dynamical CPA is an extension of the SSF to the dynamical case, and equivalent to the dynamical mean field theory in the metal–insulator transition. This equivalence is discussed in Sect. 3.6. In the last section, we extend the theory to the first-principles dynamical CPA and discuss the quantitative aspects of the finite-temperature magnetism in Fe, Co, and Ni.

3.1 Stoner Theory The Hartree–Fock–Stoner theory is useful to understand the origin of the ferromagnetism at the ground state in itinerant-electron system. The Stoner condition (2.5) shows that the ferromagnetism is caused by a balance between the energy gain due to Coulomb interaction and the band energy loss of electrons via the Fermi level, though electron correlations play a significant role in its stabilization. We will disY. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_3, © Springer-Verlag Berlin Heidelberg 2012

63

64

3 Metallic Magnetism at Finite Temperatures

cuss in this section the Stoner theory at finite temperatures to understand the nature of the mean-field theory of magnetism. Finite-temperature properties of the Stoner theory are obtained from the selfconsistent equations (2.11) for average charge n per atom and (2.12) for magnetization m per atom.  dω f (ω)ρ(ω + Δ · σ + μ), (3.1) n= σ

  m= σ dω f (ω)ρ(ω + Δ · σ + μ).

(3.2)

σ

Here we introduced the Hartree–Fock density of states (DOS) in the nonmagnetic state as ρ(ω) = ρ 0 (ω − U n/2 + εF ), the exchange potential Δ = U m/2 + h, and the change of chemical potential μ due to spin polarization. εF and h denote the Fermi level in the nonmagnetic state and the uniform magnetic field, respectively. Note that the DOS ρ(ω) satisfies the following relation by definition.   ρ(ω). (3.3) n = dω f (ω) σ

In the weak ferromagnetic limit (Δ  1), we can obtain the explicit expressions of magnetization and susceptibility in the Stoner theory as follows. Let us take the difference between (3.1) and (3.3) to obtain μ.    ρ(ω + Δ · σ + μ) − ρ(ω) = 0. (3.4) dω f (ω) σ

Expanding the above equation with respect to small Δ, we obtain μ as follows.    c 2 2 μ = aΔ 1 + a + 3b + Δ + ··· . a 2

The coefficients a, b, and c are defined as   1 1 b=− dω f (ω)ρ (2) (ω), dω f (ω)ρ (3) (ω), a=− 2a0 6a0  1 c=− dω f (ω)ρ (4) (ω). 24a0

(3.5)

(3.6)

Here ρ (n) (ω) means that ρ (n) (ω) = ∂ n ρ(ω)/∂ωn , and the coefficient a0 is defined by  (3.7) a0 = dω f (ω)ρ (1) (ω).

3.1 Stoner Theory

65

In the same way, (3.2) is expressed as     σ ρ(ω + Δ · σ + μ) − ρ(ω) . m = dω f (ω)

(3.8)

σ

Expanding the r.h.s. of the above equation with respect to small Δ and μ, and substituting the relation (3.5) into the equation, we obtain 

 m = 2a0 Δ 1 − 2a 2 + b Δ2 + · · · . (3.9) At T = 0, we have a0 = ρ, a = −ρ (1) /2ρ, and b = −ρ (2) /6ρ, where ρ, ρ (1) , and ρ (2) stand for ρ(0), ρ (1) (0), and ρ (2) (0) at T = 0, respectively. When h = 0 and T = 0, (3.9) yields the ground state magnetization,  8(ρU − 1) m(0) = . (3.10) ρU F1 U 2 Here F1 = (ρ (1) /ρ)2 − ρ (2) /3ρ. The paramagnetic susceptibility is obtained by taking the linear term in Δ at the r.h.s. of (3.9). χ(T ) =

2a0 (T ) . 1 − a0 (T )U

(3.11)

At T = 0, we have a0 = ρ, thus (3.11) reduces to (2.14), the susceptibility in the Hartree–Fock approximation at the ground state. At finite temperatures, we may adopt the low temperature expansion for the Fermi distribution function because here we are considering very weak ferromagnetism. Furthermore it should be noted that ρ(ω), the DOS in the nonmagnetic state, depends on temperature T via εF , the change of εF due to temperature. Here we express its temperature dependence explicitly as ρ(ω, T ). The change εF and ρ(ω, T ) are determined by (3.3) as εF = −

π 2 2 ρ (1) T , 6 ρ

ρ(ω, T ) = ρ(ω, 0) − ρ (1) (ω, 0)

π 2 2 ρ (1) T + ··· . 6 ρ

Then a0 (T ) defined by (3.7) is expanded as     π2 ∂f (ω) 2 ρ(ω, T ) = ρ 1 − RT + · · · . a0 (T ) = dω − ∂ω 6

(3.12) (3.13)

(3.14)

Here  R=

ρ (1) ρ

2 −

ρ (2) . ρ

(3.15)

66

3 Metallic Magnetism at Finite Temperatures

The Curie temperature is obtained from the condition that the denominator in (3.11) vanishes, a0 (TC )U = 1. With use of (3.14), it is given as  6(ρU − 1) . (3.16) TC = π 2 RρU Note that 0 < ρU − 1  1 because we are considering the very weak ferromagnetism. Therefore, we may replace ρU in the denominator of the above expression by one. Substituting (3.14) into (3.11) and using the expression of TC , we obtain χ(T )−1 =

π 2 R(ρU )2 2 T − TC2 . 12ρ

(3.17)

The inverse susceptibility in the Stoner theory is proportional to T − TC near TC , i.e., the susceptibility follows the Curie–Weiss law. But it deviates from it at higher temperatures. The magnetization at finite temperatures is obtained from (3.9):

m A + 2Bm2 + · · · = 0, (3.18) with 1 − a0 U 1 = , 4a0 2χ

1 2 2a + b U 3 . B= 32 A=

(3.19) (3.20)

Solving (3.18), we obtain  m(T ) =

−

A(T ) . 2B(T )

(3.21)

Substituting (3.17) of χ(T ) into A(T ) and replacing B(T ) with B(0) in the above expression, we reach   2 T m(T ) = m(0) 1 − . (3.22) TC Here m(0) is given by (3.10). The temperature dependence of magnetization and susceptibility in the very weak ferromagnet, which is based on the Stoner theory is summarized in (3.22) and (3.17), respectively. The change of the magnetization Δm(T ) = m(T ) − m(0) follows the T 2 law at low temperatures. In the actual transition metals, we have an additional term being proportional to T 3/2 . This is explained by the spin wave excitations as will be discussed in Sect. 4.2. Near TC , the magnetization vanishes as

3.2 Functional Integral Method Table 3.1 The LDA+Stoner theory vs. experiment of the Curie temperatures for Fe, Co, and Ni [39, 40]

67

Metal

Fe

Co

Ni

LDA+Stoner TC (K)

6000

4000

2900

Expt. TC (K)

1040

1390

630

(TC − T )1/2 . This is characteristic of the mean-field theory (see also the Ginzburg– Landau theory in Sect. 6.4). However the T 2 dependence of the inverse susceptibility in (3.17) seems to be inconsistent with the experimental data of weak ferromagnets which indicate the Curie–Weiss law in a wide range of temperature. It suggests that one has to take into account the spin fluctuations neglected in the Stoner theory. In the above analysis, we assumed a very weak ferromagnetism where 0 < ρU − 1, m  1. When the magnetization is large, we have to solve (3.1) and (3.2) self-consistently. In the realistic calculations, one can adopt the tight-binding LMTO Hamiltonian (2.167) and the first-principles Stoner parameter I (2.180) obtained by the DFT-LDA theory in Sect. 2.3.4. The Stoner model is then described by the following Hamiltonian.    1 1 † εiL − I m · σ δld niLσ + H= tiLj L aiLσ aj L σ + I m2 . (3.23) 2 4  iLσ

iLj L σ

Gunnarsson et al. [39, 40] calculated the Curie temperatures of Fe, Co, and Ni on the basis of the Stoner theory and the DFT-LDA Stoner parameters. The results are presented in Table 3.1. Calculated values are overestimated by a factor of 3–6 as compared with the experimental result. This is because the Stoner theory does not take into account the spin fluctuations at finite temperatures. The spin fluctuations reduce the magnetic energy and produce the magnetic entropy at finite temperatures. We will present the theory at finite temperatures in the following sections.

3.2 Functional Integral Method We have discussed that the Hartree–Fock type independent-particle approximation overestimates magnetic energy and underestimates magnetic entropy because it neglects charge and spin fluctuations. The approximation therefore overestimates the Curie temperatures as well as the magnetic order. Figure 3.1 shows various pictures of spin fluctuations in metallic magnetism. In the Hartree–Fock–Stoner theory (top panel), the magnetic moments on each site show no spin fluctuations with increasing temperature. The moments uniformly decrease only via Fermi distribution function due to single-particle excitations of independent electrons with exchange splitting. In the local moment system (middle panel), the amplitudes of local moments on each site are well defined, but the transverse spin fluctuations take place with increasing temperatures. In the metallic systems, in general, one can expect fluctuations in both the amplitude and the direction as shown in the bottom panel of Fig. 3.1. One

68

3 Metallic Magnetism at Finite Temperatures

Fig. 3.1 The Hartree– Fock–Stoner picture (top), local-moment picture (middle), and the spin-fluctuations expected in the metallic system (bottom)

has to explicitly take into account the spin fluctuations mentioned above to reduce the magnetic energy and to describe the magnetic entropy. The functional integral method is a suitable technique for this purpose, because auxiliary exchange fields introduced in the method can describe spin fluctuations in a simple way [41, 42]. In the functional integral method we transform the two-body interactions into a time-dependent random potential using the Hubbard–Stratonovich transformation. e

AOˆ 2

=

A π



dξ e−Aξ

2 +2AOξ ˆ

.

(3.24)

Here Oˆ is an operator, and A denotes an interaction strength. Let us consider the Hubbard model (1.51) for simplicity. We divide the Hamiltonian into two parts, the noninteracting Hamiltonian H0 and the interaction part HI , as follows. H = H0 + HI ,  † tij aiσ aj σ , H0 =

(3.25) (3.26)

i,j,σ

HI =

  (ε0 − μ)niσ + U ni↑ ni↓ . i,σ

(3.27)

i

Here ε0 and tij are the atomic level and the transfer integrals between sites i and j , respectively. U denotes the intraatomic Coulomb interaction energy parameter on each site. The chemical potential μ has been inserted for convenience. It should be noted that there are various representations of the interaction part because of the Pauli principle n2iσ = niσ . For example, ni↑ ni↓ =

1 2 ni − m2i , 4

(3.28)

3.2 Functional Integral Method

69

1 1 ni↑ ni↓ = ni − m2i , 2 2

(3.29)

or using the relation sx2 = sy2 = sz2 , we have 1 1 ni↑ ni↓ = ni − m2i . 2 6

(3.30)

Here mi = 2s i is the spin magnetic moment on site i, and mi = miz . Among various forms of interactions, we adopt the form (3.28) in the following, because in this case the simplest approximation to the functional integral method, which is called the static approximation, yields the Hatree–Fock approximation at the ground state. The free energy F is expressed in the interaction representation as follows [43].    β  HI (τ ) dτ . (3.31) e−β F = Tr e−βH0 T exp − 0

Here β denotes the inverse temperature (i.e., 1/kB T , kB being the Boltzmann constant), T denotes the time-ordered product (T product), and HI is the interaction (3.27). Note that the interactions HI (τn ) and HI (τn ) between different times commute each other under T product, because they are the Bose type operators consisting of the even products of creation and annihilation operators. Discretizing the integral with respect to time τ , we have the following relation under T product.   exp −



β

HI (τ ) dτ

0



 N N β  = exp −  (ε0 − μ)niσ (τn ) N n=1 i=1

N

×

N  n=1 i=1

 πβU 2 πβU 2 exp − n (τ ) + m (τ ) . (3.32) n n 4πN  i 4πN  i

Here N  is the number of segments of time interval [0, β]. Applying the Hubbard–Stratonivich transformation (3.24) to the above interaction terms of n2i (τn ) (m2i (τn )) at each time τn , we reach e

−β F

=

  N

 δξi (τ )δηi (τ ) Z 0 [ξ, η]

i=1



1  × exp − U 4 N



i=1 0

Here the functional integral 



β

dτ

!

ηi2 (τ ) + ξi2 (τ )

 .

(3.33)

δξ(τ ) for the auxiliary field ξ(τ ) is defined by

   N

 βU dξ(τn ) δξ(τ ) ≡ N →∞ , √  4π N n=1

(3.34)

70

3 Metallic Magnetism at Finite Temperatures

and Z 0 [ξ, η] is the partition function for noninteracting electrons under timedependent random fields ξ(τ ) and η(τ ).    Z 0 [ξ, η] = Tr T exp −

β



dτ H τ, ξ(τ ), η(τ )

 .

(3.35)

0

Note that H (τ, ξ(τ ), η(τ )) denotes the one-electron Hamiltonian in the random charge and exchange fields, which is defined by 



† H τ, ξ(τ ), η(τ ) = viσ ξi (τ ), ηi (τ ) niσ (τ ) + tij aiσ (τ )aj σ (τ ), (3.36) iσ

i,j,σ

where the time-dependent potential viσ (ξi (τ ), ηi (τ )) is defined by

1 1 viσ ξi (τ ), ηi (τ ) = ε0 − μ ∓ i U ηi (τ ) − U ξi (τ )σ. 2 2

(3.37)

Defining the energy functional E[ξ, η] by  U 1 E[ξ, η] ≡ − ln Z 0 [ξ, η] + β 4β i

 0

β

! dτ ηi2 (τ ) + ξi2 (τ ) ,

the free energy F given by (3.33) is expressed as follows.   1 F = − ln δξi δηi e−βE[ξ,η] . β

(3.38)

(3.39)

i

In order to obtain a more explicit form of the partition function Z 0 [ξ, η], we separate the time-dependent Hamiltonian (3.36) as follows. Hλ (τ, ξ, η) = H0 (τ ) + λH1 (τ, ξ, η).

(3.40)

Here we defined H0 (τ ) by the second term at the r.h.s. of (3.36), and defined H1 (τ, ξ, η) by the first term. We introduced arbitrary interaction strength parameter λ for convenience. Then the logarithmic derivative of the partition function is expressed by a Green function Gλij σ (τ, τ  ) as follows.  ∂ ln Zλ0 =− ∂λ j,σ

 0

β



dτ vj σ (τ )Gλjj σ τ, τ + ,

(3.41)

where τ + means τ plus an infinitesimal positive number. The Green function for the time-dependent potentials viσ (τ ) is defined by β

Tr(T aiσ (τ )aj†σ (τ  )e− 0 Hλ (τ

Gλij σ τ, τ  ≡ − β   Tr(T e− 0 Hλ (τ ) dτ )

 ) dτ 

) .

(3.42)

3.2 Functional Integral Method

71

The Green function is obtained by solving the following Dyson equation [43, 44].



Gλij σ τ, τ  = gij σ τ − τ   β 





dτ  gikσ τ − τ  λvkσ τ  Gkj σ τ  , τ  . (3.43) + 0

k

τ )

Here gij σ (τ − is the Green function for noninteracting Hamiltonian H0 . The Dyson equation is solved in the matrix form as follows.

−1 (3.44) Gλ = g −1 − λv , where (Gλ )iτ σj τ  σ  = Gλij σ (τ, τ  )δσ σ  . Substituting the expression into (3.41) and integrating it with respect to λ from 0 to 1, we obtain ln Z 0 = ln Z00 + Sp ln(1 − vg).

(3.45)

Here Z00 is the partition function for the noninteracting system H0 , and Sp means a trace over site, time, and spin. Substituting (3.45) into (3.38), we obtain

1 E[ξ, η] = − ln Z00 + Sp ln g + Sp ln g −1 − v β  U  β ! + dτ ηi2 (τ ) + ξi2 (τ ) . (3.46) 4β 0 i

Equations (3.39) and (3.46) indicate that the statistical mechanics of two-body interaction have been transformed into those of classical integrals of determinants with infinite dimensions. Note that the Green function (3.42) for the time-dependent dynamical potentials viσ (τ ) differs from the temperature Green function for the interacting Hamiltonian (3.25). The latter is defined by

   † (3.47) Giσj σ  τ − τ  = − T aHiσ (τ )aHj σ τ . † (τ )) is the Heisenberg representation of the annihilation (creHere aHiσ (τ ) (aHiσ ation) operator of an electron on site i with spin σ ; aHiσ (τ ) = eτ K aiσ e−τ K † † −τ K (aHiσ (τ ) = eτ K aiσ e ), where K = H − μN , μ being the chemical potential. We can prove that the temperature Green function is given by the Green function (3.42) as   δξi δηi Giσj σ  (τ, τ  )e−βE[ξ,η]



 i Giσj σ  τ − τ  = Giσj σ  τ, τ  =   . (3.48) δξi δηi e−βE[ξ,η] i

To prove the above relation, we express the temperature Green function (3.47) with use of the interaction representation as follows.

β  

e−β F Giσj σ  τ − τ  = − Tr T aiσ (τ )aj†σ  τ  e− 0 HI (τ ) dτ . (3.49)

72

3 Metallic Magnetism at Finite Temperatures

Since terms with different times in the integral of HI (τ  ) at the r.h.s. commute each other under T product, we can apply the Hubbard–Stratonovich transformation (3.24) discretizing the integral as (3.32), so that we obtain the formula (3.48). The charge and magnetic moment on site i are obtained by taking the derivatives of (3.39) with respect to the atomic level εi0 and the local magnetic field hi on site i as ∂F /∂εi0 and −∂F /∂hi .   1  β

+ dτ Giiσ τ, τ , (3.50) ni  = β 0 σ   1  β

mi  = σ dτ Giiσ τ, τ + . (3.51) β 0 σ The average ∼ at the r.h.s. of (3.50) and (3.51) is defined by a thermal average with respect of the energy functional E[ξ, η].   δξi δηi (∼)e−βE[ξ,η] ∼ ≡

i  

. δξi δηi e−βE[ξ,η]

(3.52)

i

We can obtain alternative expressions with use of the auxiliary fields. To derive the expressions, we write the local charge as ni  = ∂E[ξ, η]/∂εi0 . Since εi0 appears in the energy E[ξ, η] via the time-dependent energy εi (τ ) ≡ εi0 − μ ∓ U iηi (τ )/2 according to (3.38), we take the derivative for every time segment τn as   ∂ E[ξ, η] . (3.53) ni  = ∂εi (τn ) n We can rewrite ∂E/∂εi (τn ) in the above equation using ∂E/∂ηi (τn ) via the relation ∂E/∂ηi (τn ) = ∓i(U/2)∂E/∂εi (τn ) + (Δτ/2β)U ηi (τn ). By integration by parts, we obtain an alternative expression as follows. ni  = ±iηi .

(3.54)

For the local magnetic moment, we make use of the time-dependent magnetic field hi (τn ) ≡ U ξi (τn )/2 + hi instead of the time-dependent energy εi (τn ). Taking the same steps, we obtain the following formula. mi  = ξi .

(3.55)

Here ξi and ηi are the time-averaged static field variables defined by   1 β 1 β ξi = ξi (τ ) dτ, ηi = ηi (τ ) dτ. (3.56) β 0 β 0 The results (3.54) and (3.55) indicate that the averages of the zero frequency components of the charge and exchange fields determine local charge and magnetization on the same site. In particular, (3.55) suggests that a ‘magnetic moment’ ξi on site i thermally fluctuates as shown in the bottom panel of Fig. 3.1.

3.2 Functional Integral Method

73

In the functional integral method, one has to evaluate the energy functional E[ξ, η] (i.e., (3.46)) first, and then perform the functional integrals as found in (3.39) and (3.52). In order to obtain the free energy describing spin fluctuations, we make use of the time-averaged static variables (3.56). Using the identity 

  1 β 1 = dξi δ ξi − ξi (τ ) dτ β 0

      βU βU 1 β dξi dxi exp −2πixi ξi (τ ) dτ , (3.57) ξi − = 4π 4π β 0 

one can express the free energy (3.32) by means of the static variables ξi and ηi as follows.   N    N  βU e−β F = dξi dηi δξi (τ )δηi (τ ) dxi dyi Z 0 [ξ, η] 4π i=1

i=1

N  √   β  

1 βπU β 2 × exp − U ξi (τ ) dτ + ixi ξi (τ ) − ξi dτ 4 β 0 0 i=1

N  √   β  

1 βπU β 2 − U ηi (τ ) dτ + iyi ηi (τ ) − ηi dτ . × exp 4 β 0 0 i=1

(3.58) The simplest approximation is to replace the time-dependent fields ξi (τ ) and ηi (τ ) in the Hamiltonian of Z 0 [ξ, η] with the static ones, i.e., ξi and ηi , as H (τ, ξ(τ ), η(τ )) → H (τ, ξ, η). This is called the static approximation. The partition function (3.35) then reduces to the following form.  

Zst0 (ξ, η) = Tr exp −βHst (ξ, η) ,

(3.59)

and Hst (ξ, η) =

 i

viσ (ξi , ηi )niσ +



† tij aiσ aj σ .

(3.60)

i,j,σ

Here viσ (ξi , ηi ) is a random static potential (3.37) in which the dynamical-field variables ξi (τ ) and ηi (τ ) have been replaced by the static ones. 1 1 viσ (ξi , ηi ) = ε0 − μ ∓ iU ηi − U ξi σ. 2 2

(3.61)

The remaining functional integrals can be performed with use of the Gaussian integrals, and we obtain the following expression for the free energy Fst in the

74

3 Metallic Magnetism at Finite Temperatures

static approximation. e

−β Fst

=

  N βU i=1

4π

dξi

   N  2

βU 1 0 2 dηi Zst (ξ, η) exp − βU ηi + ξi . 4π 4 i=1

(3.62) Here Zst0 (ξ, η) is the partition function (3.59) for the Hamiltonian Hst with the random static potential viσ (ξi , ηi ). The free energy (3.62) is also written as follows.

  βU βU 1 dξi dηi e−βEst (ξ,η) , (3.63) Fst = − ln β 4π 4π i

N

1  2

Est (ξ, η) = −β −1 ln Tr e−βHst (ξ,η) + U ηi + ξi2 . 4

(3.64)

i=1

Needless to say, the static approximation is exact in the noninteracting limit. It is also exact in the atomic limit because the Hamiltonian H0 commutes with the interaction HI there, so that the replacement H (τ, ξ(τ ), η(τ )) → H (τ, ξ, η) is justified. Also, the approximation becomes exact in the high temperature limit where the time interval [0, β] becomes zero. At T = 0, on the other hand, one can take the saddle point of Est (ξ, η) to calculate the free energy. The ground state energy is then given by

1  ∗ 2 H  = μN + Hst (ξ ∗ , η∗ )0 + U ηi + ξi∗ 2 . 4

(3.65)

i

Here (∼)0 means the average with respect to the Hamiltonian Hst (ξ ∗ , η∗ ) at T = 0. The saddle point values ξi∗ and ηi∗ are determined from the conditions, ∂Est /∂ξi = 0 and ∂Est /∂ηi = 0, i.e.,



ξi∗ = mi 0 ξ ∗ , η∗ , ∓iηi∗ = ni 0 ξ ∗ , η∗ . (3.66) These coupled equations for ξi∗ and ηi∗ are nothing but the Hartree–Fock equations, and the potential viσ (ξ ∗ , η∗ ) reduces to the Hartree–Fock one, ε0 − μ + U ni 0 /2 − U mi 0 σ/2. Therefore, the energy (3.65) is identical with the Hartree–Fock energy at the ground state.

3.3 Single-Site Theory in the Static Approximation The static approximation to the functional integral method yields a physical picture of spin fluctuations as indicated in Fig. 3.1, though it is a rather crude approximation at low temperatures. In this section we present a single-site theory of spin

3.3 Single-Site Theory in the Static Approximation

75

fluctuations based on the static approximation, which is known as the single-site spin fluctuation theory (SSF) [45–49]. We adopt here the saddle point approximation to the charge fields, ζi ≡ ∓iηi∗ = ni 0 (ξ, ±iζ ), and express the free energy (3.63) and the energy potential (3.64) as follows.   βU 1 dξi e−βEst (ξ ) , (3.67) Fst = − ln β 4π i 

 1  2  1 Est (ξ ) = dω f (ω) Im Tr ln(z − Hst ) + U ξi − ζi2 . (3.68) π 4 i

Here z = ω + iδ, δ being the infinitesimal positive number. Hst is the one-electron Hamiltonian matrix defined by   (3.69) (Hst )iσj σ  = viσ (ξ )δij + tij (1 − δij ) δσ σ  . The random static potential is now given by 1 1 viσ (ξ ) = ε0 − μ + U ζi (ξ ) − U ξi σ, 2 2

(3.70)

and ζi (ξ ) = ni 0 (ξ ) denotes the Hartree–Fock charge on site i when the random exchange fields {ξj } are given. The Hamiltonian matrix Hst in (3.68) consists of the random potential (v)iσj σ  = viσ δij δσ σ  and the transfer integral matrix (t)iσj σ  = tij δσ σ  ; Hst = v + t. In order to reduce the number of integrations in the free energy (3.67), we introduce an energydependent complex potential called the coherent potential Σσ (z) into the potential part as Hst = Σ + t + δv. δv denotes a scattering potential from a medium Σ; δv = v − Σ. The energy (3.68) is then expressed as follows.    

1 ˜ ˜ + 1U Est (ξ ) = F + dω f (ω) Im Tr ln(1 − δv G) ξi2 − ζi2 . (3.71) π 4 i

Here F˜ is the coherent part of the free energy given by    1 ˜ F = dω f (ω) Im Tr ln(z − Σ − t) . π

(3.72)

The second term at the r.h.s. of (3.71) is a contribution from the scattering potential ˜ is defined by δv. The coherent Green function G   ˜ iσj σ  = (z − Σ − t)−1 (G) δ δ . (3.73) iσj σ ij σ σ The last term of the r.h.s. of (3.71) is the Gaussian contribution to the free energy. In order to expand the second term in (3.71) with respect to sites, we divide the

76

3 Metallic Magnetism at Finite Temperatures

˜ into the diagonal part F and the off-diagonal F  as follows. Green function G ˜ = F + F , G

(3.74)

where   (F )iσj σ  = Fiσ (z)δij δσ σ  = (z − Σ − t)−1 iσ iσ δij δσ σ  , 

˜ iσj σ (1 − δij )δσ σ  . F iσj σ  = (G)

(3.75) (3.76)

We can then expand the energy (3.71) with respect to the site as follows. E(ξ ) = F˜ +



Ei (ξi ) + E(ξ ).

(3.77)

i

The second term at the r.h.s. of (3.77) consists of the single-site energies on each site.  

1 1 Ei (ξi ) = dω f (ω) Im ln 1 − δviσ (z, ξ )Fiσ (z) + U ξi2 − ζi2 . (3.78) π 4 σ Here δviσ (z, ξ ) = viσ (ξ ) − Σσ (z). The last term in (3.77) describes the inter-site interaction.  

 1 E(ξ ) = dω f (ω) Im Tr ln 1 − t˜F  . π

(3.79)

(3.80)

Here t˜ is the single-site t matrix for the scattering δv. t˜ = (1 − δvF )−1 δv.

(3.81)

In the single-site approximation (SSA), we neglect the last term in (3.77), and obtain the free energy as follows. FSSA = F˜ − β −1

 i

 ln

βU dξi e−βEi (ξi ) . 4π

(3.82)

The coherent potential Σσ (z) is determined so that the contribution from the inter-site corrections (3.80) becomes as small as possible. This implies that  t˜iσ  =

 δviσ = 0. 1 − δviσ Fiσ

(3.83)

3.3 Single-Site Theory in the Static Approximation

77

Here the average ∼ means a classical average with respect to the impurity energy Ei (ξi ).  βU dξi (∼)e−βEi (ξi ) 4π ∼ =  . (3.84) βU −βEi (ξi ) dξi e 4π Equation (3.83) is equivalent to the following condition (see Appendix A). (i)

Giσ (z, ξ ) = Fiσ (z).

(3.85)

Here (i)

Giσ (z, ξ ) =

Fiσ

(z)−1

1 . − δviσ (z, ξ )

(3.86)

(i)

The function Giσ (z, ξ ) is an on-site Green function for a system with an impurity potential viσ embedded in an effective medium Σσ (z), whose Hamiltonian is given by (i)

  H (z) j lσ = viσ (z, ξ )δij + Σσ (z)(1 − δij ) δj l + tj l (1 − δj l ). (3.87) The coherent Green function Fiσ (z) is defined by (3.75), and expressed by the density of states ρ(ε) for the noninteracting Hamiltonian tij as  ρ(ε) dε . (3.88) Fiσ (z) = z − Σσ (z) − ε The form is known as the Lehmann representation of the Green function. The single-site approximation with use of a site-diagonal effective potential Σσ (z) is known as the coherent potential approximation (CPA), and (3.83) and (3.85) are called the CPA equation. These equations are also verified to be equivalent to the stationary condition of the free energy (3.82) with respect to the coherent potential (see Appendix A). δFSSA = 0. δΣσ (z)

(3.89)

The CPA was first introduced as a method to treat the electronic structure of disordered alloys [50, 51], which will be discussed in more detail in Sect. 8.2. The schematic picture of the CPA equation is given in Fig. 3.2. We have replaced the random potentials {vj σ } of the one-electron Hamiltonian Hst (i.e., (3.69)) with the coherent potential Σσ (z) at the surrounding sites in order to reduce the number of field variables. We then have an impurity system described by an effective Hamiltonian (3.87), and obtain the impurity Green function (3.86). The l.h.s. in Fig. 3.2 shows the thermal average of such a system. On the other hand, we can consider a coherent system in which all the random potentials are replaced by the coherent

78

3 Metallic Magnetism at Finite Temperatures

Fig. 3.2 Physical picture for the coherent potential approximation (CPA). The left-hand side (l.h.s.) shows an impurity system with a random potential viσ on site i, which is embedded in the effective potential Σσ . The right-hand side shows a uniform system with the effective potential only. When the average is taken at the l.h.s. with respect to a random potential viσ , it should be identical with the r.h.s

ones, as shown at the r.h.s. in Fig. 3.2. We then obtain the coherent Green function (3.88). The CPA equation (3.85) indicates that these states should be identical when the effective medium is chosen to be best. One can rederive the formula (3.54) of local charge on site i from the single-site free energy (3.82) by using the stationary conditions (3.89) and ∂Ei (ξi )/∂ζi = 0, as well as the relations ni  = ∂FSSA /∂εi0 .   ni  = ζi (ξ ) ,   ζi (ξ ) = dω f (ω) ρiσ (ω, ξ ).

(3.90) (3.91)

σ

Here ρiσ (ω, ξ ) is the single-site density of states (DOS) given by ρiσ (ω, ξ ) = −

1 (i) Im Giσ (z, ξ ). π

(3.92)

In the same way, we can rederive the formula (3.55) of the local magnetic moment on site i using the relation mi  = −∂FSSA /∂hi . mi  = ξi .

(3.93)

The average at the r.h.s. is now taken with respect to the single-site energy Ei (ξi ) (see (3.84)). Equation (3.93) indicates that the magnetic moment at finite temperatures is determined by a thermal average of a flexible local moment ξi with respect to the energy potential Ei (ξ ). The amplitude of local moment is obtained from the relations m2i  = ni  − 2ni↑ ni↓  and ni↑ ni↓  = ∂FSSA /∂Ui [52]. m2i  = ni  −

  1 2 2 2 ζi (ξ ) − ξi  + . 2 βU

(3.94)

3.3 Single-Site Theory in the Static Approximation

79

The second term at the r.h.s. of the equation shows that the amplitude of local moment is flexible and can vary with elevated temperature. The entropy S per atom is also obtained from the relation S = −∂FSSA /∂T /L, where L denotes the number of atoms [52].  



 S = − dω ρ(ω) ˜ 1 − f (ω) ln 1 − f (ω) + f (ω) ln f (ω)  + ln

βU 1 dξ e−β(Ei (ξ )−Ei (ξ )) − . 4π 2

(3.95)

Here ρ(ω) ˜ is the DOS per atom for the coherent state, which is given by ρ(ω) ˜ = −π −1 Im σ Fiσ (z). The first term expresses an entropy for the independent particle system. This is essentially the same as in the Stoner theory. The second term causes a magnetic entropy associated with the local √ magnetic moment ξ . The last term is a constant which originates in the prefactor βU/4π of the integral. The energy per atom in the SSA is obtained from the relation H −μN  = FSSA +β −1 S as  H  = μn +

  1 2 2 2 . dω f (ω)ωρ(ω) ˜ − U ζi (ξ ) − ξi  + 4 βU

(3.96)

Here n is the electron number per atom. The energy in the static approximation has the same form as the Hartree–Fock one (2.13), though the DOS ρ(ω) ˜ and the amplitude ξi2  shows a strong temperature dependence. It should be noted that the coherent potential Σσ (z) and the chemical potential μ are self-consistently determined in the single-site theory of spin fluctuations by means of the CPA equation (3.85) and the condition (3.90) for a given charge n. The noninteracting DOS ρ(ε) for the Hamiltonian matrix tij is assumed to be given in the theory. We first assume Σσ and μ, and calculate the coherent Green function (3.88). Then we obtain the Hartree–Fock charge ζi (ξ ) for each ξ solving the self-consistent equations (3.70) and (3.91). Using these solutions, we can calculate the energy potential Ei (ξ ) according to (3.78), therefore the l.h.s. of the CPA equation (3.85) taking the thermal average with respect to Ei (ξ ). When the CPA equation (3.85) is not satisfied, we may renew the coherent potential, for example, using the following equation.  Σσ(new) (z) = Σσ(old) (z) −

(i)

Giσ (z, ξ ) − Fiσ (z) (i)

Giσ (z, ξ )Fiσ (z)

.

(3.97)

old

The above equation was found in the averaged t-matrix approximation to the CPA (new) [51] (see also Appendix B). The renewed potential Σσ (z) improves the selfconsistency, and after several iterations we may find the CPA solution Σσ (z) for each z. Next we calculate the average charge according to (3.90). When the obtained charge is not consistent with the given charge n, we can renew the chemical potential μ, for example, by using Newton’s method, and return to the beginning of

80

3 Metallic Magnetism at Finite Temperatures

Fig. 3.3 Energy potential curves Ei (ξ ) = Ei (ξ ) − Ei (0) calculated at half-filling for U/W = 0.5, 1.0, 1.5, 2.0 in the paramagnetic state at T /W = 0.06 √ [53], where W denotes a half the band width. Semi-elliptical model density of states ρ(ε) = 2 W 2 − ε 2 /πW 2 is assumed for noninteracting electrons

the procedure mentioned above. When the self-consistency for both Σσ (z) and μ is achieved, we can calculate various physical quantities. The basic behavior of local magnetic moments is determined by the singlesite energy Ei (ξ ) (see (3.78)). Let us consider the paramagnetic state at halffilling, where ζi (ξ ) = 1 irrespective of ξ . When ξ is small, we have Ei (ξ ) = Ei (0) + Aξ 2 /2. For small U , the Gaussian term in Ei (ξ ) (i.e., the second term at the r.h.s. of (3.78)) is dominant because it is proportional to U , so that A > 0, while A < 0 for large U because the kinetic energy term (i.e., the first term at the r.h.s. of (3.78)) decreases as ξ increases and its magnitude is proportional to U 2 . When ξ is large, the Gaussian part again becomes dominant. Therefore Ei (ξ ) shows a single minimum structure for small U , while it shows a double-minimum structure for large U as shown in Fig. 3.3. When the system shows a single minimum structure in energy potential Ei (ξ ) (see the curve for U/W = 0.5 in Fig. 3.3), the magnetic moment ξ vanishes at T = 0. In this case, it may appear with the polarization of the effective medium. The magnetic moment fluctuates around the minimum point with elevating temperature. When the system shows a double-minimum structure (see the curves for U/W = 1.5 and U/W = 2.0 in Fig. 3.3), the magnetic moment given by the condition ∂Ei (ξi )/∂ξi = 0 appears at T = 0, and thermally fluctuates between the two minima. In this case, large spin fluctuations which changes their direction are possible. Accordingly, the magnetic entropy of ln 2 occurs. We call such a system the local moment system. Various model calculations for the ferromagnetism have been performed with use of the single-site theory in the static approximation. In the single-site theory, the ferromagnetism is defined as a polarized state of the effective medium;

3.3 Single-Site Theory in the Static Approximation

81

Fig. 3.4 Energy potential curves Ei (ξ ) = Ei (ξ ) − Ei (0) calculated at a temperature below (solid curve) and above (dashed curve) the Curie temperature for n = 1.44 and 2U/W = 2.29 (‘bcc Fe’) [54], W being a width of the model band shown in the inset

Σ↑ (z) = Σ↓ (z). The ferromagnetic solution is usually found in the system with non-half-filled bands and a large Coulomb interaction. Figure 3.4 shows a numerical example for electron number n = 1.44 which corresponds to bcc Fe when electron number n is multiplied by 5 (number of d orbitals) [54]. In the paramagnetic state, we have double minima in energy curve Ei (ξ ). Below the Curie temperature TC , we have an energy-dependent molecular field Σ↑ (z) − Σ↓ (z), consequently the energy potential curve becomes asymmetric as shown by the solid curve in the figure, and the ferromagnetism with the magnetization at the minimum is stabilized. Figure 3.5 shows another example of the energy potential curves for a system with n = 1.8 on the fcc lattice, which might correspond to fcc Ni. In this case, we find the single minimum structure both above and below TC . With the appearance of the energy-dependent molecular field Σ↑ (z) − Σ↓ (z), the minimum point shifts to the positive region of ξ , and the ferromagnetism is realized. Temperature dependence of calculated magnetic moments is presented in Fig. 3.6. In the case of n = 1.44 and 2U/W = 2.29 (‘bcc Fe’), the ground-state magnetization is m0 = 0.511 (= 2.56/5) μB . With increasing temperature the magnetization decreases and vanishes at TC /W = 0.0253 (TC = 900 K for W = 0.45 Ry). In the case of n = 1.80 and 2U/W = 3.43 (‘fcc Ni’), we obtain the ground-state magnetization m0 = 0.146 (= 0.73/5) μB and the Curie temperature TC /W = 0.0192 (TC = 530 K for W = 0.35 Ry). In both cases, we find the paramagnetic susceptibilities which follow the Curie–Weiss law, χ = (m2eff /3)/(T − TC ). Here the effective Bohr magneton numbers are obtained as meff = 1.75m0 for ‘bcc Fe’ and meff = 3.20m0 for ‘fcc Ni’, respectively, while the experimental data indicate that

82

3 Metallic Magnetism at Finite Temperatures

Fig. 3.5 Energy potential curves Ei (ξ ) calculated below (solid curve) and above (dashed curve) the Curie temperature for n = 1.80 and 2U/W = 3.43 (‘fcc Ni’) [54]. The inset shows the model density of states for fcc Ni

Fig. 3.6 Magnetization vs. temperature curves, inverse susceptibilities, amplitudes of local moments for ‘bcc Fe’: n = 1.44, 2U/W = 2.29 (solid curves), and for ‘fcc Ni’: n = 1.80, 2U/W = 3.43 (dotted curves) [54]

meff = 1.44m0 for bcc Fe and meff = 2.68m0 for fcc Ni. These results show that the SSF qualitatively explains the metallic ferromagnetism, and significantly improve the Hartree–Fock results.

3.4 Dynamical CPA Theory

83

3.4 Dynamical CPA Theory The single-site theory of spin fluctuations (SSF) discussed in the last section is based on the static approximation, i.e., the high-temperature approximation. The theory reduces to the Hartree–Fock approximation at the ground state and thus it does not take into account electron correlations at zero temperature. More important is that it does not describe the Fermi liquid state as found in metals and alloys at low temperatures. It is also expected that the dynamical spin fluctuations which are not taken into account further reduce the magnetization and the Curie temperature. In order to overcome these difficulties, we extend here the theory to the dynamical case [54–57]. The theory is called the dynamical coherent-potential approximation (CPA). We introduced the energy dependent effective medium Σσ (z) to the static energy potential (3.68) in the SSF. In the dynamical CPA, we introduce a time-dependent effective medium Σ defined by

(3.98) (Σ)iτ σj τ  σ  ≡ Σiσ τ − τ  δij δσ σ  , and insert it into the potential part of the energy functional (3.46) as follows.



˜ −1 + Sp ln(1 − δv G). ˜ (3.99) Sp ln g −1 − v = Sp ln g −1 − Σ − δv = Sp ln G Here δv = v − Σ, and the coherent Green function is defined by 

 ˜ iτ σj τ  σ  = g −1 − Σ −1 δ . (G) iτ σj τ  σ σ σ

(3.100)

The energy functional E[ξ, η] (see (3.46)) is then divided into the coherent part and the remaining correction term as  1 ˜ ˜ ln Z0 + Sp ln(1 − δv G) β  U  β

dτ ηi2 (τ ) + ξi2 (τ ) . + 4β 0

E[ξ, η] = −

(3.101)

i

The partition function Z˜ 0 is defined by ln Z˜ 0 = ln Z00 + Sp ln(1 − Σg). Defining the diagonal coherent Green function F and the off-diagonal F  as

(F )iτ σj τ  σ 

˜ = F + F , G 

−1  ˜ iτ σ iτ  σ δij δσ σ  , = g −1 − Σ = (G) iτ σ iτ  σ  

˜ iτ σj τ  σ (1 − δij )δσ σ  , F iτ σj τ  σ  = (G)

(3.102) (3.103) (3.104)

one can expand the scattering potential δv in the energy functional with respect to sites as follows.  E[ξ, η] = F˜ [Σ] + Ei [ξi , ηi ] + E[ξ, η]. (3.105) i

84

3 Metallic Magnetism at Finite Temperatures

Here the first term at the r.h.s. of the above equation is the coherent part of the free energy defined by F˜ [Σ] = −β −1 ln Z˜0 = −β −1 ln Z00 − β −1 Sp ln(1 − Σg).

(3.106)

The impurity contribution to the energy functional is given by  β

U −1 Ei [ξi , ηi ] = −β tr ln(1 − δvi Fi ) + dτ ηi2 (τ ) + ξi2 (τ ) , (3.107) 4β 0 where tr means a trace over time and spin, and we used the notation (Fi )τ σ τ  σ  ≡ (F )iτ σ iτ  σ  is used. The last term in (3.105) is the off-diagonal contribution to the energy functional, which is given by

E[ξ, η] = −β −1 Sp ln 1 − t˜F  . (3.108) Here t˜ is the single-site t matrix defined by t˜ = (1 − δvF )−1 δv.

(3.109)

Substituting (3.105) into (3.39), we obtain the following expression of the free energy. 1   1  ln δξi δηi e−βEi [ξi ,ηi ] − ln e−βE 0 . F = F˜ [Σ] − (3.110) β β i

Here ∼0 means the single-site average with respect to its energy functionals.    δξi δηi (∼)e−β i Ei [ξi ,ηi ] ∼0 ≡

i  

δξi δηi e

−β

 i

.

(3.111)

Ei [ξi ,ηi ]

i

The single-site approximation (SSA) is to neglect the inter-site term (the last term) in the free energy expansion (3.110). We choose the effective medium so that the corrections from the last term becomes minimum. As expected from (3.108), the condition is given by t˜i 0 = 0; that is, t˜i  = 0. Here ∼ stands for the single-site average as  δξi δηi (∼)e−βEi [ξi ,ηi ] ∼ =  . δξi δηi e−βEi [ξi ,ηi ]

(3.112)

(3.113)

The matrix t˜i in (3.112) denotes the single-site t matrix due to a local potential δvi . t˜i = (1 − δvi Fi )−1 δvi ,

(3.114)

3.4 Dynamical CPA Theory

85

and

(δvi )τ σ τ  σ  = τ δviσ τ, τ  τ δσ σ 



= τ viσ (τ )δ τ − τ  − Σiσ τ − τ  τ δσ σ  .

(3.115)

Here τ = β/N , N being the number of mesh in the time interval [0, β]. Equation (3.112) means that the single-site t matrix due to the scattering potential from the effective medium should vanishes in average. One can rewrite the condition (3.112) as (Fi−1 − δvi )−1  = Fi .

(3.116)

Equations (3.112) and (3.116) are referred as the dynamical CPA equation. The r.h.s. of (3.116) is the diagonal part of the coherent Green function, i.e., (3.103). The l.h.s. is an impurity Green function for the system with impurity potential vi embedded in the effective medium Σ . One can verify the fact as follows. The Hamiltonian in the interaction representation is given by H (i) (τ ) = H˜ (τ ) +

 σ

β 0



!

† dτ  aiσ (τ ) viσ (τ )δ τ − τ  − Σiσ τ − τ  aiσ τ  . (3.117)

Here H˜ (τ ) is the time-dependent Hamiltonian for the coherent potential. H˜ (τ ) = H0 (τ ) +

 iσ

β

0





† dτ  aiσ (τ )Σiσ τ − τ  aiσ τ  .

(3.118)

The impurity Green function for the Hamiltonian H (i) (τ ) is defined as follows (see (3.42)).

Tr(T (i) Gj lσ τ, τ  ≡ −

† aj σ (τ )alσ (τ  )e−

Tr(T e−

β 0

β 0

H (i) (τ  ) dτ 

H (i) (τ  ) dτ 

)

.

(3.119)

)

The Dyson equation for the Green function Gj lσ (τ, τ  ) is given by [43, 44] (i)

(i) Gj lσ τ, τ 



= gj lσ τ − τ  +



β

dτ1 dτ2



0

gj kσ (τ − τ1 )

k

!

 × Σkσ (τ1 − τ2 ) + δviσ (τ1 , τ2 )δik G(i) klσ τ2 , τ . (3.120) The above equation yields the following solution in the matrix representation.

−1 G(i) = Fi−1 − δvi .

(3.121)

86

3 Metallic Magnetism at Finite Temperatures

This is identical with the inside of the average at the l.h.s. of (3.116). Therefore the CPA equation (3.116) is expressed as G(i)  = Fi .

(3.122)

The dynamical CPA equations (3.112) and (3.122) have the same form as (3.83) and (3.85) in the static approximation. Thus the physical picture of the dynamical CPA equation is the same as given in Fig. 3.2. We have replaced the dynamical potentials {vj σ } with dynamical coherent potential Σ at the surrounding sites in order to reduce the number of field variables. We have then an impurity system described by an effective Hamiltonian (3.117), thereby obtaining the impurity Green function (3.121). The l.h.s. in Fig. 3.2 shows such a system. On the other hand, we can consider a coherent system in which all dynamical potentials have been replaced by coherent ones as shown at the r.h.s. in Fig. 3.2. Corresponding Green function is the coherent Green function (3.103). The CPA equation (3.122) indicates that these states should be identical when the effective medium is chosen to be best. The free energy in the dynamical CPA is obtained from (3.110) as follows. 1  ln δξi δηi e−βEi [ξi ,ηi ] . (3.123) FCPA = F˜ [Σ] − β i

One can verify that the dynamical CPA equations (3.112) and (3.122) are also equivalent to the following stationary condition for the CPA free energy. δFCPA = 0. δΣiσ (τ − τ  )

(3.124)

In order to verify the above relation, we express the functional derivative of δFCPA /δΣnσ (τ − τ  ) as     δ δFCPA ˜ = F+ Ei [ξi , ηi ] . (3.125) δΣnσ (τ − τ  ) δΣnσ (τ − τ  ) i

Substituting (3.106) and (3.107) into (3.125), we obtain    δFCPA −1 −(Fn )τ  τ σ + Fn (1 − δvn Fn )−1 τ  τ σ = −β  δΣnσ (τ − τ )    δFi −1 . − tr (1 − δvi Fi ) δvi δΣnσ (τ − τ  )

(3.126)

i

Here tr means taking the trace over time and spin. The first and the second terms at the r.h.s. of the above equation are written by the single-site t-matrix on site n, so that we obtain     δFi δFCPA ˜ ˜ = Fn tn Fn τ  τ σ − . (3.127) tr ti  δΣnσ (τ − τ  ) δΣnσ (τ − τ  ) i

3.4 Dynamical CPA Theory

87

Using the CPA condition (3.112), we reach (3.124). The formulae (3.50), (3.51), (3.54), and (3.55) of the local charge and magnetic moment on site i can be rederived by taking the derivatives ∂FCPA /∂εi0 and −∂FCPA /∂hi , after having introduced a site-dependent atomic level εi0 and a magnetic field hi acting on site i. The Green function Giiσ (τ, τ + ) in (3.50) and (3.51), however, should be replaced by the coherent Fiiσ (τ, τ + ). 1 β

dτ Fiiσ τ, τ + , β 0 σ  1 β

mi  = σ dτ Fiiσ τ, τ + . β 0 σ ni  =

(3.128)

(3.129)

In actual calculations of physical quantities, the Fourier representation is more convenient. The Green function for noninteracting Hamiltonian H0 is expressed as

1  gij σ (iωl )e−iωl (τ −τ ) . gij σ τ − τ  = β

(3.130)

l

Here ωl is the Matsubara frequency for the anti-periodic function, ωl = (2l +1)π/β. The Fourier transform gij σ (iωl ) is obtained as gij σ (iωl ) =

 i|kk|j  k

iωl − εk

.

(3.131)

In the same way, the Fourier representations of the coherent Green function and the coherent potential are given as follows.

1  Fiσ (iωl )e−iωl (τ −τ ) , Fiσ τ − τ  = β

(3.132)



1  Σiσ τ − τ  = Σiσ (iωl )e−iωl (τ −τ ) . β

(3.133)

l

l

Here Fiσ (iωl ) is given by (3.88) with z = iωl and Σiσ (iωl ) should be determined by the dynamical CPA equation. For the field variables associated with the dynamical potential, it is suitable to adopt the Fourier transform as the periodic function (i.e., v(τ < 0) = v(τ + β)).  ξi (τ ) = ξi (iωl )e−iωl τ , (3.134) l

where ωl = 2lπ/β. The dynamical potential is therefore expressed as  vi (τ ) = vi (iωl )e−iωl τ . l

(3.135)

88

3 Metallic Magnetism at Finite Temperatures

Here the Fourier representation of the potential is given by ! 1 vj σ (iωl ) = (ε0 − μ)δl0 − U ±iηj (iωl ) + ξj (iωl )σ . 2

(3.136)

The free energy in the Fourier representation has the same form as (3.123): 1  FCPA = F˜ [Σ] − ln δξi δηi e−βEi [ξi ,ηi ] . (3.137) β i

The coherent part F˜ [Σ] defined by (3.106) is now given by

1 1 F˜ [Σ] = − ln tr e−βH0 − Sp ln(1 − Σg) β β ! 1  ln iωl − εk − Σσ (iωl ) . =− β

(3.138)

k,l,σ

Here Sp stands for a trace over site, frequency, and spin. The functional integral in (3.137) is given by  ∞    βU βU 2 δξ = (3.139) dξ(0) d ξ(iωl ) , 4π 2π l=1

where d 2 ξ(iωl ) = d Re ξ(iωl ) d Im ξ(iωl ). The energy functional defined by (3.107) is expressed in the frequency representation as follows. 2  2 ! 1 1   ηi (iωl ) + ξi (iωl ) . (3.140) Ei [ξi , ηi ] = − tr ln(1 − δvi Fi ) + U β 4 l

Here tr stands for a trace over frequency and spin. It should be noted that δvi is not diagonal in the frequency representation because it contains the Fourier transform of the time dependent potential. (δvi )ilσj mσ  = δviσ (iωl , iωm )δij δσ σ  , δviσ (iωl , iωm ) = viσ (iωl − iωm ) − Σiσ (iωl )δlm .

(3.141) (3.142)

The Fourier transform of the dynamical CPA equation (3.122) is given by G(i) iσ (iωl , iωl ) = Fiσ (iωl ),

−1   −1 . G(i) iσ (iωl , iωl ) = Fi − δvi lσ lσ

(3.143) (3.144)

The stationary condition (3.124), which is equivalent to the dynamical CPA equation, is expressed in the frequency representation as follows. δFCPA = 0. δΣiσ (iωl )

(3.145)

3.4 Dynamical CPA Theory

89

The dynamical CPA is also obtained from the Dyson equation (3.43) for a timedependent Green function by making use of the multiple scattering theory (see Appendix B). The dynamical CPA completely takes into account the single-site spin and charge fluctuations. Thus the theory is the best single-site approximation. As will be shown in Sect. 3.6, the dynamical CPA is equivalent to the many-body CPA in the disordered alloys [61], the dynamical mean-field theory (DMFT) in the metal– insulator transition [62, 63], and the projection operator method CPA in the excitation problem [64]. When we approximate the dynamical potential with the static one: viσ (iωl − iωm ) ≈ viσ (0)δlm , we can perform the integration with respect to the dynamical variables {ξ(iωl ), η(iωl )}(l = 0) using the Gaussian integrals. The free energy (3.137) then reduces to

  βU βU −1 ˜ dξi dηi e−βEi (ξi ,ηi ) , (3.146) FCPA = F [Σ] − β ln 4π 4π i

and

1

Ei (ξi , ηi ) = −β −1 tr ln 1 − δvi (0)Fi + U ξi2 + ηi2 . 4

(3.147)

Here (δvi (0))lσ mσ  = (viσ (0) − Σσ (iωl ))δlm δσ σ  . ξi = ξi (0) and ηi = ηi (0) are the static field variables defined by (3.56). When all the sites are equivalent to each other one can omit the subscript i. Making use of the saddle-point approximation to the charge field (∂Ei /∂ηi = 0), we obtain the free energy per atom in the static approximation as follows.  βU −1 ˜ FCPA = F [Σ] − β ln dξ e−βEi (ξ ) . (3.148) 4π Here FCPA (F˜ ) now denotes the free energy (coherent free energy) per site, and the static energy potential is expressed as follows. Ei (ξ ) = −β −1



1

ln 1 − δvσ (iωl , ξ )Fσ (iωl ) + U ξ 2 − ζ 2 (ξ ) . (3.149) 4 lσ

Here δvσ (iωl , ξ ) is defined by (3.79), and ζ (ξ ) =

1  (i) Giσ (iωl , ξ ). β

(3.150)

lσ

(i)

The static Green function Giσ (z, ξ ) is defined by (3.86). In the same approximation, the dynamical CPA equation (3.143) reduces to the static one as (i)

Giσ (iωl , ξ ) = Fiσ (iωl ).

(3.151)

90

3 Metallic Magnetism at Finite Temperatures

The frequency sums in (3.149) and (3.150) can be transformed into the integrals on the real axis of the complex z plane, so that one can verify that the free energy (3.148), the energy potential (3.149), the self-consistent equation for the charge field (3.150) and the CPA equation (3.151) are identical with those obtained in the singlesite spin fluctuation theory (SSF) presented in the last section (see (3.82), (3.78), (3.91), and (3.85)). That is, the dynamical CPA reduces to the SSF when the static approximation is made. The derivation is given in Appendix C.

3.5 Dynamical CPA with Harmonic Approximation We have seen in Sect. 3.3 that the single-site theory based on the static approximation (i.e., SSF) describes well the high temperature properties of metallic magnetism. Thus it is reasonable to take into account the dynamical corrections starting from the static limit, i.e., from the high temperature limit. In this section we present such an approach based on the harmonic approximation [54]. We assume that all the sites are equivalent, and express the free energy (3.137) as follows.  βU −1 ˜ dξ e−βEeff (ξ ) . (3.152) FCPA = F − β ln 4π Here we omitted the site indices for simplicity. FCPA (F˜ ) denotes the free energy (coherent part of the free energy) per site. The effective potential Eeff (ξ ) projected onto the static field variable ξ is defined by    ∞ βU βU d 2 ξ(iωl ) d 2 η(iωl ) e−βE[ξ,η] . (3.153) e−βEeff (ξ ) = 2π 2π l=1

Note that we have taken the saddle point η∗ for static charge field η = η(0) for simplicity. In order to start from the static limit, we divide the dynamical potential (3.136) ˜ where into the static part v0 and the dynamical part v˜ as v = v0 + v, (v0 )lσ mσ  ≡ viσ (0)δlm δσ σ  ,

(v) ˜ lσ mσ  ≡ viσ (iωl − iωm ) − viσ (0)δlm δσ σ  .

(3.154) (3.155)

Accordingly, the first term in the single-site energy functional (3.140) can be divided into two parts as follows. tr ln(1 − δvF ) = tr ln(1 − δv0 F ) + tr ln(1 − v˜ g). ˜

(3.156)

Here δv0 = v0 − Σ , and the static Green function g˜ is defined by (g) ˜ lσ mσ  ≡

−1   −1 g − v0 = g˜ σ (iωl )δlm δσ σ  , lσ mσ 

(3.157)

3.5 Dynamical CPA with Harmonic Approximation

91



−1 g˜ σ (iωl ) = Fσ (iωl )−1 − δvσ (iωl , ξ ) .

(3.158)

Here δvσ (iωl , ξ ) is defined by (3.79). Note that the diagonal part g˜ σ (iωl ) is identical with (3.86) with z = iωl . Substituting the single-site energy functional (3.140) with the relation (3.156) into (3.153), we can divide the effective potential into two parts as follows. Eeff (ξ ) = Est (ξ ) + Edyn (ξ ).

(3.159)

The first term Est (ξ ) is the static energy potential given by (3.149). The second term is the dynamical contribution Edyn (ξ ) given by    ∞ βU 2 βU 2 −βEdyn (ξ ) d ξ(iωl ) d η(iωl ) D↑ D↓ e = 2π 2π l=1

− ×e

∞ βU  (|ξ(iωl )|2 + η(iωl )|2 ) 2 l=1

.

(3.160)

Here we used the relation tr ln(1 − v˜ g) ˜ = ln det(1 − v˜ g). ˜ The determinant Dσ is defined by   Dσ = det δlm − v˜σ (iωl − iωm )g˜ σ (iωm ) . (3.161) The r.h.s. of (3.160) is a Gaussian average of D↑ D↓ with respect to the dynamical variables {ξ(iωl ), η(iωl )}. Thus we express the dynamical contribution as e−βEdyn (ξ ) = D↑ D↓ .

(3.162)

Here the upper bar denotes the Gaussian average. It should be noted that the dynamical CPA equation (3.143) is obtained from the stationary condition (3.145) and the free energy (3.152). Making use of the stationary condition, we find that the dynamical Green function (3.144) is obtained from the dynamical part of the effective potential as follows.   δEdyn (ξ ) G(i) (iω , iω ) = g ˜ (iω ) − β . (3.163) l l σ l σ κσ (iωl )δΣσ (iωl ) eff Here κσ (iωl ) = 1 − Fσ (iωl )−2 δFσ (iωl )/δΣσ (iωl ) and the average ∼eff at the r.h.s. denotes a classical average with respect to the effective potential Eeff (ξ ). In this reformulation of the dynamical CPA with the use of the effective potential, the problem reduces to how to obtain the dynamical contribution Edyn (ξ ) in (3.162) and (3.163). In order to calculate the dynamical contribution, we expand here the determinant with respect to the frequency modes of the dynamical potential v˜σ (iων ) as follows.   (Dνσ − 1) + (Dνν  σ − Dνσ − Dν  σ + 1) + · · · , (3.164) Dσ = 1 + ν

(ν,ν  )

92

3 Metallic Magnetism at Finite Temperatures

  Dνσ = det δlm − v˜σ (iων )δl−m,ν + v˜σ (iω−ν )δl−m,−ν g˜ σ (iωm ) , 

Dνν  σ = det δlm − v˜σ (iων )δl−m,ν + v˜σ (iω−ν )δl−m,−ν g˜ σ (iωm )

 − v˜σ (iων  )δl−m,ν  + v˜σ (iω−ν  )δl−m,−ν  g˜ σ (iωm ) .

(3.165)

(3.166)

The first term in (3.164) corresponds to the zeroth approximation (i.e. the static approximation), the second term expresses the independent scattering due to each dynamical potential vσ (iων ), and the higher order terms express the mode-mode couplings. We neglect here the mode-mode coupling terms and only take into account the independent frequency terms. This is known as the harmonic approximation (HA) [58, 59]. The approximation is exact up to the second order of U in the weak Coulomb interaction region. Numerical studies also indicate that it describes well the strong Coulomb interaction limit [60]. The approximation is independent of the Coulomb interaction strength, so that the HA is considered to be suitable for description of the intermediate region. The determinant Dνσ can be expanded with respect to the dynamical potential as Dνσ =

 n ∞ 

n i 1 (n) 4β v˜σ (ν)v˜σ (−ν) Bνσ . n! 8πν

(3.167)

n=0

(n)

Here we adopted a simplified notation v˜σ (ν) = vσ (iων ). Bνσ consists of a linear combination of 2n products of the static Green function (see Appendix D). Because D νσ = 1 and Dνσ Dν  σ  = D νσ D ν  σ  = 1 for ν  = ν, we obtain from (3.162) the dynamical part of effective potential in the harmonic approximation as   ∞  1 (Dν↑ Dν↓ − 1) . (3.168) Edyn (ξ, η) = − ln 1 + β ν=1

Here Dν↑ Dν↓ is calculated by using the Gaussian integrals as follows. Dν↑ Dν↓ =

∞ 

 U 2l

l=0

i 2πν

2l

(l)

(l)

(3.169)

Bν↑ Bν↓ .

Substituting (3.168) into (3.163), we obtain the impurity Green function as



"

 G(i) σ (iωl , iωl ) = g˜ σ (iωl ) +

∞  ν=1

δ(Dν↑ Dν↓ ) # κσ (iωl )δΣσ (iωl )

∞  1+ (Dν↑ Dν↓ − 1)

.

(3.170)

eff

ν=1

In the actual calculations of Dν↑ Dν↓ , we make use of the exact expression up to 2l-th order. For higher orders, we adopt an asymptotic approximation, which is

3.5 Dynamical CPA with Harmonic Approximation

93

exact in the high frequency limit [54]. Dν↑ Dν↓ =

l  n=0

 U

2n

i 2πν

2n

(n) (n) Bν↑ Bν↓

+

∞ 

 U

2n

n=l+1

i 2πν

2n

(n) (n) B˜ ν↑ B˜ ν↓ . (3.171)

Here B˜ νσ denotes the coefficient Bνσ in the asymptotic approximation. The local charge and moment in the harmonic approximation are derived from the free energy (3.152) as   n = ζ (ξ ) eff , (3.172) (n)

(n)

m = ξ eff .

(3.173)

The above expressions are consistent with the general formula, (3.54) and (3.55). Note that ζ (ξ ), the electron number under static exchange field ξ , is not defined by (3.150) now. Instead it is given by the impurity Green function for the dynamical potential (3.170) as ζ (ξ ) =

1  (i) Gσ (iωl , iωl ). β

(3.174)

lσ

The expression of the amplitude of local moment is obtained with use of the relation ni↑ ni↓  = ∂FCPA /∂U .      2  ∂Edyn (ξ ) 1  2 2 1 ξ eff − −2 . (3.175) m = n − ζ (ξ )2 eff + 2 2 βU ∂U v eff Here [∼]v means to take the derivative fixing the static potential vσ (0). The harmonic approximation (HA) takes into account the dynamical contributions successively starting from the high temperature limit (i.e., the static limit). Figure 3.7 shows examples of effective potentials in the paramagnetic ‘bcc Fe’, which are calculated by means of the dynamical CPA + HA and the same model DOS as in the static case (see Fig. 3.4). The potential curve in the static approximation shows a double-minimum structure, and hardly depends on the temperature. The dynamical corrections are small at high temperatures (see the dot-dashed curve), so that the effective potential is close to the static one. The dynamical corrections increase with decreasing temperature, and become significant near the Curie temperature TC as shown by dotted curve, so that the effective potential shows a single minimum at ξ = 0, indicating a disappearance of ‘local moment’. The same feature is obtained below TC . This indicates that the thermodynamics of the metallic ferromagnets in the intermediate region of the Coulomb interaction are dominated by the quantum fluctuations rather than the thermal spin fluctuations, though the orbital degeneracy plays an important role in the real system as will be discussed in Sect. 3.7. The dynamical effects strongly influence the magnetic properties. Figure 3.8 shows the result of model calculations for temperature variation of magnetic moments and susceptibility for ‘bcc Fe’. Both the static and dynamical CPA lead to the

94

3 Metallic Magnetism at Finite Temperatures

Fig. 3.7 Effective potentials of the paramagnetic ‘bcc Fe’ calculated with use of the dynamical CPA + HA and the model DOS given in Fig. 3.4 [54]. W denotes the d band width for noninteracting system. The static contribution (dotted curve) has double minima. The dynamical correction is small at T /TC = 24 (dot-dashed curve), while it is significant at T /TC = 2.4 (thin dotted curve), so that the total effective potential (solid curve) shows a single minimum structure

Fig. 3.8 Magnetization vs. temperature curves, inverse susceptibility curves, and the amplitude of local moment of ‘bcc Fe’ calculated with use of the model DOS given in Fig. 3.4 [54]. Solid curves (dotted curves): the results obtained by the dynamical CPA + HA (the static approximation)

Curie–Weiss susceptibility. The Curie temperature, however, is reduced by a factor of two due to dynamical spin and charge fluctuations.

3.6 Dynamical CPA and Dynamical Mean-Field Theory

95

3.6 Dynamical CPA and Dynamical Mean-Field Theory The dynamical CPA presented in Sects. 3.4 and 3.5 is equivalent to some singlesite theories of electron correlations. We clarify in this section the basic ideas of these theories, i.e., the many-body CPA (MB-CPA) in the disordered alloys [61], the dynamical mean-field theory (DMFT) in the metal–insulator transition [62, 63], and the projection operator method CPA (PM-CPA) in the excitation problem [64], and verify that the dynamical CPA is equivalent to these theories.

3.6.1 The Many-Body CPA and Its Equivalence to the Dynamical CPA The many-body CPA (MB-CPA) is an extension of the CPA in disordered alloys to the correlated electron system. The theory has been applied to the Ni-based alloys to elucidate the electron-correlation effects on magnetism [61]. To clarify the relation of the MB-CPA to the dynamical CPA, we consider here a pure metal and adopt the Hubbard model with an intra-atomic Coulomb interaction. The theory starts from the temperature Green function in the interaction representation as 

 †  − 0β H (τ  ) dτ   τ e . Giσ τ − τ  = −Z −1 Tr T aiσ (τ )aiσ

(3.176)

Here Z denotes the partition function of the system. The Hamiltonian H (τ ) in the interaction representation is approximated by an effective Hamiltonian H˜ (τ ), i.e., (3.118) with the time-dependent coherent potential Σiσ (τ − τ  ):  β



† ˜ dτ  aiσ (τ )Σiσ τ − τ  aiσ τ  . (3.177) H (τ ) = H0 (τ ) + iσ

0

 Note that we defined here the noninteracting Hamiltonian as H0 = iσ εσ0 niσ +  † tij aiσ aj σ , and εσ0 = ε 0 − μ, so that the interaction is given by HI = ij σ  ˜ i U ni↑ ni↓ . The diagonal Green function Fiσ (τ − τ ) for H (τ ) is given as follows according to the Dyson equation.



−1  Fiσ τ − τ  = g −1 − Σ . iτ σ iτ  σ

(3.178)

Here gij σ (τ − τ  ) is the Green function for the noninteracting Hamiltonian H0 . To find the best coherent potential, we consider an impurity Hamiltonian embedded in the effective medium as follows.  β  †



(i) ˜ dτ  aiσ (τ )Σiσ τ − τ  aiσ τ  + U ni↑ (τ )ni↓ (τ ). H (τ ) = H (τ ) − 0

σ

(3.179)

96

3 Metallic Magnetism at Finite Temperatures

Fig. 3.9 Schematic picture showing the many-body CPA. The left-hand side shows an impurity with a real Coulomb interaction Ui on site i and the surrounding effective potential Σσ . The right-hand side shows a uniform system with the effective potential only

Here the coherent potential on site i has been replaced by the real Coulomb interaction U ni↑ (τ )ni↓ (τ ). Note that (3.179) corresponds to (3.117) in the dynamical CPA, but the time-dependent random potential viσ (τ ) has been replaced by the real interaction HI (τ ) in the MB-CPA (see Fig. 3.9). The on-site Green function for the impurity system is given by 

 (i) †  − 0β H (i) (τ  ) dτ   τ e . Giσ τ − τ  = −Zi−1 Tr T aiσ (τ )aiσ

(3.180)

Here Zi denotes the partition function of the system. The coherent potential in the many-body CPA is determined so that the diagonal impurity Green function agrees with the coherent Green function (see Fig. 3.9).



(i) (3.181) Giσ τ − τ  = Fiσ τ − τ  . (i) (τ − τ  ) has to be obtained separately by Note that the impurity Green function Giσ using one of the many-body techniques. In order to clarify the equivalence of the MB-CPA to the dynamical CPA, we discretize the integral of H (i) (τ  ) in (3.180) as in (3.32), and apply the Hubbard– Stratonovich transformation (3.24) directly to the impurity Green function (3.180). We obtain then the following relation.

(i) (i) (3.182) Giσ τ − τ  = Giσ (ξ, η, τ, τ  ).  Here G(i) iσ (ξ, η, τ, τ ) is identical with the diagonal part of the time-dependent Green function (3.119) in the dynamical CPA. The above relation indicates that the CPA equation (3.181) in the many-body CPA is identical with the CPA equation (3.122) in the dynamical CPA. Thus the many-body CPA is equivalent to the dynamical CPA.

3.6.2 The DMFT and Its Equivalence to the Dynamical CPA The dynamical mean field theory (DMFT) was developed to clarify the metal– insulator transition in infinite dimensions, and has extensively been applied to various problems in strongly correlated electron systems [62, 63]. It is equivalent to the

3.6 Dynamical CPA and Dynamical Mean-Field Theory

97

MB-CPA thus to the dynamical CPA as well. In order to introduce the DMFT and to prove its equivalence to the MB-CPA, we employ the temperature Green function which is expressed by the path integral method as follows [65].  

∗ Giσ τ − τ  = −Z −1 Daj∗σ Daj σ e−S aiσ (τ )aiσ (τ ), (3.183) jσ

 S=

β

dτ



0

iσ

  ∗

∂ ∗ − μ aiσ (τ ) + H {a }{a} . aiσ (τ ) ∂τ

(3.184)

Here Z denotes the partition function to the action S. The Hamiltonian operator H ({a ∗ }{a}) in the action S is defined by the same form as the original Hamiltonian ∗ and a in the interaction representation, though aiσ iσ are now the Grassmann variables being conjugate to the creation and annihilation operators on the same site. $N  ∗ Da (= ∗ Daiσ iσ n=1 daiσ (τn )daiσ (τn )) denotes the path integrals for these variables. It is well-known that the Feynman diagram rule obtained in the path integral formulation is exactly the same as in the usual Green function technique [65], so that we can derive the same Dyson equation as follows.



Gij σ τ, τ  = gij σ τ − τ   β  β 

+ dτ1 dτ2 gikσ (τ − τ1 )Σklσ (τ1 − τ2 )Glj σ τ2 , τ  . 0

0

kl

(3.185) Here gij σ (τ − τ  ) is the Green function for the noninteracting Hamiltonian H0 . Σij σ (τ − τ  ) denotes the self-energy which should be obtained from the Feynman diagrams. Furthermore, we note that the same Dyson equation (3.185) is also obtained from the following effective action according to the diagrammatic technique.     β

∂ ∗ S = − μ aiσ (τ ) + H  {a ∗ }{a} , dτ aiσ (τ ) (3.186) ∂τ 0 iσ



H  {a ∗ }{a} = H0 {a ∗ (τ )}{a(τ )}  β



∗ + dτ  aiσ (τ )Σij σ τ − τ  aj σ τ  . (3.187) ij σ

0

The DMFT has been developed in infinite dimensions [62]. There,√the hopping integrals |tij | on the hyper-cubic lattice for example are scaled as |t|/ 2d, so that the band width becomes finite for any dimensions d. Accordingly, the √ off-diagonal Green function in the noninteracting system becomes of order of 1/ 2d. Since the Feynman diagrams to the off-diagonal self-energy have at least three electron internal lines between the different sites, the contribution from all the off-diagonal

98

3 Metallic Magnetism at Finite Temperatures

√ self-energy diagrams to the Dyson equation becomes of order of 2d(1/ 2d)3 = √ 1/ 2d. Therefore the contribution from the off-diagonal self-energy Σij σ (iωl ) (i = j ) vanishes in the limit d = ∞ [66]. The DMFT determines the site-diagonal self-energy (or the momentum independent self-energy) Σσ (iωl ) by using the following diagonal Green function in the path integral representation.  



(i) ∗ ∗ τ , (3.188) Daiσ Daiσ e−Si aiσ (τ )aiσ Giσ τ − τ  = −Z (i)−1 



β

Si = −

dτ 0

 +

0

σ β

dτ 







∗ aiσ (τ ) F (i)−1 σ τ − τ  aiσ τ 

σ

β

(3.189)

dτ U ni↑ (τ )ni↓ (τ ). 0

∗ (τ )a (τ ). S is an impurity action with an effective field and Here niσ (τ ) = aiσ iσ i the local Coulomb interaction. Z (i) denotes the partition function to the action Si . (i) ∗ Da  Daiσ iσ denotes the path integrals for these variables. Fσ (τ − τ ) in (3.189) is called the Weiss-field function. According to the Dyson equation of the Green function (3.188), it is given in the Fourier representation as follows.

Fσ(i) (iωl )−1 = Σσ (iωl ) + Giσ (iωl )−1 . (i)

(3.190)

(i)

Since the site-diagonal Green function Giσ (iωl ) should be equal to the average of the Green function of the momentum representation (1/(iωl − εσ0 − Σσ (iωl ) − εk )), we have the relation,  ρ(ε) dε (i) , (3.191) Giσ (iωl ) = iωl − εσ0 − Σσ (iωl ) − ε where ρ(ε) is the one-electron density of states for tij . This means that   iωl − εσ0 − Σσ (iωl ) = R Gσ(i) (iωl ) .

(3.192)

HereR denotes the reciprocal function to the Hilbert transform (i.e., x = R[y] when y = dε ρ(ε)/(x − ε)). Eliminating Σσ (iωl ) from (3.190) and (3.192), we obtain the expression of the Weiss function by means of the impurity Green function as   Fσ(i) (iωl )−1 = iωl − εσ0 + Gσ(i) (iωl )−1 − R Gσ(i) (iωl ) .

(3.193)

Equations (3.188), (3.189), and (3.193) form the self-consistent equations in the DMFT. Note that the Green function (3.188) for an impurity action has to be solved by using one of the many-body theories (e.g., the quantum Monte-Carlo method [62] and the numerical renormalization group method [67]).

3.6 Dynamical CPA and Dynamical Mean-Field Theory

99

In order to clarify the relation between the DMFT and the MB-CPA, we note that the Green function (3.183) and the Hamiltonian H ({a ∗ }{a}) in the action S, i.e., (3.184), have the same form as those in the interaction representation in the Fock space (see (3.176)). Thus we can construct the MB-CPA using the path integral method taking the same steps as in the original MB-CPA. To realize it, we approximate the self-energy Σij σ (τ − τ  ) in (3.187) by the diagonal Σiσ (τ − τ  ), and consider the following coherent action with the effective Hamiltonian H˜ ({a ∗ }{a}). S˜ =



β

dτ 0





∗ aiσ (τ )

iσ

 ∗

∂ ˜ − μ aiσ (τ ) + H {a }{a} , ∂τ



 H˜ {a ∗ }{a} = H0 {a ∗ (τ )}{a(τ )} +

 0

iσ

β

(3.194)





∗ dτ  aiσ (τ )Σiσ τ − τ  aiσ τ  . (3.195)

Note that the Hamiltonian H˜ ({a ∗ }{a}) is the same as (3.177) in which the creation and the annihilation operators have been replaced by their conjugate variables ∗ (τ ) and a (τ ). aiσ iσ The action S˜ describes the r.h.s. of Fig. 3.9, and its Green function is given by  



Fiσ τ − τ  = −Z˜ −1



˜ ∗ τ , Daj∗σ Daj σ e−S aiσ (τ )aiσ

(3.196)

jσ

˜ The Green function Fiσ (τ − τ  ) is where Z˜ is the partition function to action S. identical to the one obtained from the Hamiltonian (3.177) because both satisfy the same Dyson equation (3.178). The impurity action describing the l.h.s. of Fig. 3.9 is expressed as  S

(i)

=

dτ 0

H

(i)

β





∗ aiσ (τ )

iσ

 ∗

∂ (i) − μ aiσ (τ ) + H {a }{a} , ∂τ

∗



{a }{a} = H˜ {a ∗ }{a} −



β

dτ 

0



(3.197)





∗ aiσ (τ )Σiσ τ − τ  aiσ τ 

σ

∗ ∗ (τ )ai↓ (τ )ai↓ (τ )ai↑ (τ ). + Ui ai↑

(3.198)

The impurity Green function of effective action S (i) is given by

(i) Giσ τ − τ  = −Z (i)−1

  jσ



(i) ∗ τ , (3.199) Daj∗σ Daj σ e−S aiσ (τ )aiσ

100

3 Metallic Magnetism at Finite Temperatures

where Z (i) is the partition function to action S (i) . The impurity Green function satisfies the following Dyson equation.



(i) (i) Giσ τ − τ  = Fiσ τ − τ   β  β

(i) (i) + dτ1 dτ2 Fiσ (τ − τ1 )Λiσ (τ1 − τ2 )Giσ τ2 − τ  . 0

0

(3.200) Here Λiσ (τ − τ  ) is the self-energy for the impurity Hamiltonian (3.198). (i) The cavity Green function Fiσ (τ − τ  ) in (3.200) is defined by

(i) Fiσ τ − τ  = −Z˜ (i)−1

 



˜ (i) ∗ τ , Daj∗σ Daj σ e−S aiσ (τ )aiσ

(3.201)

jσ

S˜ (i) =



β

dτ 0

 iσ

 

∂ ∗ − μ aiσ (τ ) + H˜ (i) {a ∗ }{a} . aiσ (τ ) ∂τ

(3.202)

Here Z˜ (i) is the partition function for action S˜ (i) , and H˜ (i) ({a ∗ }{a}) is the cavity Hamiltonian defined by the first two terms at the r.h.s. of (3.198). The Green function satisfies the Dyson equation.



(i) τ − τ  = Fiσ τ − τ  Fiσ  β  − dτ1 0

0

β

(i) dτ2 Fiσ (τ − τ1 )Σiσ (τ1 − τ2 )Fiσ τ2 − τ  . (3.203)

In the Fourier representation it is expressed as Fiσ (iωl )−1 = Fiσ (iωl )−1 + Σiσ (iωl ). (i)

(3.204)

The CPA equation corresponding to Fig. 3.9 is then given by



(i) Giσ τ − τ  = Fiσ τ − τ  .

(3.205)

The impurity Green function in (3.199) is identical with that of the original MBCPA, i.e., (3.180), because the path integral method to the Green function leads to the same Feynman diagram scheme as in the usual Green function method, and the same unperturbed Green function and the same Coulomb matrix elements appear there. This means that the CPA equation (3.205) derived by the path integral formulation is identical with the CPA equation (3.181) in the MB-CPA. Let us rewrite the self-consistent equations for the MB-CPA based on the path (i) integral method. The impurity Green function Giσ (τ − τ  ) given by (3.199) can also

3.6 Dynamical CPA and Dynamical Mean-Field Theory

101

be expressed by an effective local action Si as follows.  



(i)  (i)−1 ∗ ∗ Daiσ Daiσ e−Si aiσ (τ )aiσ Giσ τ − τ = −Z  τ , σ

Z (i) =

 

∗ Daiσ Daiσ e−Si .

(3.206) (3.207)

σ

Here Si is defined by e

−Si

=

  

Daj∗σ Daj σ



e−S . (i)

(3.208)

j =i σ

Alternatively, Si = S˜ (i) −



β

dτ



0



∗ aiσ (τ )Σiσ





τ − τ  aiσ τ  +

˜ (i)

=

β

dτ Ui ni↑ (τ )ni↓ (τ ), 0

σ

e−S



  

˜ Daj∗σ Daj σ e−S .

(3.209) (3.210)

j =i σ

The path integral at the r.h.s. of (3.210) can be performed exactly by using the Gaussian formula after the Fourier transform of the Grassmann variables, so that we obtain an explicit form of the local effective action Si as Si = −

  1 ∗ aiσ (iωl ) Fiσ (iωl )−1 + Σiσ (iωl ) aiσ (iωl ) β lσ



β

+

dτ Ui ni↑ (τ )ni↓ (τ ).

(3.211)

0

Here we have omitted the constant term, which is canceled by the same factor in Z (i) in (3.206). Using the Dyson equation (3.204), the action (3.211) reduces to the following form.  β  β 



(i)−1 ∗ Si = − τ − τ  aiσ τ  dτ dτ  aiσ (τ ) Fi σσ 0

 +

0

σ

β

dτ Ui ni↑ (τ )ni↓ (τ ).

(3.212)

0 (i)−1

Here (Fi

)σ σ (τ − τ  ) is defined by (i)−1

1  (i)   Fi τ − τ = Fiσ (iωl )−1 e−iωl (τ −τ ) . σσ β l

(3.213)

102

3 Metallic Magnetism at Finite Temperatures

When the coherent potential is site-independent and all the sites are crystallographically equivalent, the frequency representation of the CPA equation (3.205) can be simplified as Gσ(i) (iωl ) = Fσ (iωl ).

(3.214)

Here the subscript i has been omitted for simplicity. The r.h.s. of (3.214) is given by  ρ(ε) dε . (3.215) Fσ (iωl ) = iωl − εσ0 − Σσ (iωl ) − ε Introducing the reciprocal function R[F ] of the Hilbert transform, we can rewrite (3.215) as   (3.216) Σσ (iωl ) = iωl − εσ0 − R Gσ(i) (iωl ) . From (3.204), (3.214), and (3.216), we can express the cavity Green function by (i) means of the impurity Green function Gσ (iωl ) as follows.   (i) Fiσ (iωl )−1 = iωl − εσ0 + Gσ(i) (iωl )−1 − R Gσ(i) (iωl ) . (3.217) Equations (3.206), (3.212), and (3.217) form the self-consistent equations in the MB-CPA, and are identical with (3.188), (3.189), and (3.193) in the DMFT. Thus we have verified the equivalence between the DMFT and the MB-CPA.

3.6.3 The Projection Operator Method CPA and Summary of Relations The MB-CPA and DMFT are based on the temperature Green function. These approaches often require the numerical analytic continuation on the complex energy plane for the calculation of the single-particle excitation spectra. The projection operator method CPA (PM-CPA) [64] is a single-site theory based on the retarded Green function, and is equivalent to the theories mentioned above.  The retarded Green function GR iσj σ  (t − t ) is defined by



    †   aHiσ (t), aHj . GR iσj σ  t − t = −iθ t − t σ t +

(3.218)

† (t) (aHiσ (t)) is the creation (annihilation) operHere θ (t) is the step function, aHiσ † † −iH t ator in the Heisenberg representation which is defined by aHiσ (t) = eiH t aiσ e iH t −iH t (aHiσ (t) = e aiσ e ), and [ , ]+ denotes the anticomutator between the Fermion operators. The average ∼ is taken over the grand canonical ensemble. Note that the Heisenberg representation of operator A is expressed by means of the Liouville operator L as A(t) = exp(iLt)A. The Liouville operator L is a super-operator defined by LA = [H, A]− , where [ , ]− is the commutator between operators.

3.6 Dynamical CPA and Dynamical Mean-Field Theory

103

The PM-CPA starts from the Fourier transform of the retarded Green function with the projection technique [68]. By making use of a Laplace transform, the Fourier transform of (3.218) is expressed by an inner-product in the operator space as follows.    1 †  † . (3.219) a (z) = a GR ij σ iσ  z − L jσ Here z = ω + iδ with δ being an infinitesimal positive number. The inner product between the operators A and B is defined by (A|B) = [A+ , B]+ . A basic idea of the PM-CPA is to approximate the Liouville operator by means of ˜ an energy dependent effective Liouville operator L(z). It is defined for an operator A as   ˜ (3.220) L(z)A = H˜ (z), A − ,  H˜ (z) = H0 + Σσ (z)niσ . (3.221) iσ

Here Σσ (z) is a coherent potential or a single-site self-energy. The coherent potential is determined as follows. We introduce a Liouville operator L(i) (z) for an impurity system such that   L(i) (z)A = H (i) (z), A − , (3.222)  H (i) (z) = H˜ (z) − Σσ (z)niσ + U ni↑ ni↓ . (3.223) σ

According to the equation of motion we obtain the diagonal Green function for the Liouville operator L(i) (z) as

−1 . Gσ(i) (z) = Fσ (z)−1 − Λ(i) σ (z) + Σσ (z)

(3.224)

Here Fσ (z) is the diagonal coherent Green function defined by (3.215) in which iωl (i) has been replaced by z. Λσ (z) is the self-energy for the impurity system, consisting (i) of the Hartree–Fock potential and the reduced memory function Gσ (z). (i)

2 Λ(i) σ (z) = U ni−σ  + U Gσ (z), 



−1 (i) (i) Gσ (z) = A†iσ  z − L (z) A†iσ .

(3.225) (3.226)

Note that the operator space of the memory function has been expanded from {|aj†σ )} to {|A†j σ ) = |aj†σ δnj −σ )}, where δnj −σ = nj −σ − nj −σ . The Liouville operator (i)

(i)

L (z) is defined by L (z) = QL(i) (z)Q with use of the projection operator Q =  (i) 1 − P and P = j |aj†σ )(aj†σ |. L (z) operates on the operator space orthogonal to the original space {|aj†σ )}.

104

3 Metallic Magnetism at Finite Temperatures

Fig. 3.10 Schematic diagram showing the equivalence among the many-body CPA (MB-CPA), the dynamical CPA (Dyn. CPA), the dynamical mean-field theory (DMFT), and the projection operator method CPA (PM-CPA)

The coherent potential or the energy dependent Liouville operator is determined from the CPA equation (see Fig. 3.9) Gσ(i) (z) = Fσ (z),

(3.227)

(i)

or equivalently Λσ (z) = Σσ (z). The CPA equation (3.227) in the PM-CPA has the same form as the one obtained from an analytic continuation of the temperature Green function in the MB-CPA (see (3.214)). Thus it is essentially the same as the MB-CPA when a suitable single-site approximation has been made for the static averages in the self-energy (3.225). (Note that the projection method treats the dynamics and the static averages independently.) The PM-CPA is extended to the nonlocal case [69]. Figure 3.10 summarizes the relation among various single-site theories. The dynamical CPA, the MB-CPA in disordered alloys, and the DMFT in the metal– insulator transition are equivalent. The single-site spin fluctuation theory (SSF) [45– 49] is obtained from the dynamical CPA as a high-temperature approximation, and the variational approach (VA) [52], which makes use of an effective potential obtained by a variational energy at low temperatures, is an adiabatic approximation to the dynamical CPA. The three equivalent theories are based on the temperature Green function. The PM-CPA in the excitation problem is based on the retarded Green function, and also equivalent to them. These relations imply that the singlesite theories developed in the magnetism and those developed in the strongly correlated electron system are unified.

3.7 First-Principles Dynamical CPA and Metallic Magnetism The dynamical CPA theory presented in Sects. 3.4 and 3.5 is based on the singleband Hubbard model. We have to take into account the realistic band structure and associated inter-orbital Coulomb interactions in order to compare theoretical results with experimental data and to discuss quantitative aspects of the theory. In the case

3.7 First-Principles Dynamical CPA and Metallic Magnetism

105

of transition metals and their alloys, electrons on the five d orbitals are responsible for the magnetism. The orbital degeneracy increases the degree of freedom on electron motion and causes different types of Coulomb interactions. Especially, the intraatomic exchange interaction parallels the spins on an atom and tends to form an atomic magnetic moment as well as associated magnetic entropy. The transverse spin fluctuations as shown in Fig. 3.1 are also realized. We introduce in this section a realistic model Hamiltonian obtained by the first-principles local density approximation (LDA) band theory [70], and outline the dynamical CPA applied to the realistic Hamiltonian [71, 72]. We will also discuss the ferromagnetism of Fe, Co, and Ni at finite temperatures on the basis of the realistic theory. We apply here the following tight-binding model Hamiltonian. H = H0 + HI .

(3.228)

The Hamiltonian for noninteracting system H0 is given by H0 =

 iLσ

0 εiL nˆ iLσ +

 iLj L σ

† 0 tiLj L aiLσ aj L σ .

(3.229)

0 is the atomic level on site i and orbital L(= lm) for the noninteracting Here εiL 0  † system. tiLj L is the transfer integral between iL and j L . aiLσ (aiLσ ) is the creation (annihilation) operator for an electron with orbital L and spin σ on site i, and nˆ iLσ = † aiLσ is a charge density operator. We added the hat sign on the charge and spin aiLσ density operators in this section in order to distinguish these operators with those in the density functional theory (DFT). In the transition metal system, for example, the 3d electrons form narrow bands as compared with the 4s and 4p electron bands. The 4s–4p electrons behave like free electrons, and may screen the Coulomb interactions between 3d electrons. Therefore we may apply the following interaction HI consisting of the intra-atomic Coulomb interactions between d electrons.    1 U1 − J nˆ ilm nˆ ilm HI = U0 nˆ ilm↑ nˆ ilm↓ + 2 m i m>m  (3.230) − J sˆ ilm · sˆ ilm . m>m

Here U0 (U1 ) and J are the intra-orbital (inter-orbital) Coulomb interaction and the exchange interaction, respectively. nˆ ilm (ˆs ilm ) with l = 2 is the charge (spin) density operator for d electrons on site i and orbital m, which is defined by nˆ ilm = σ nˆ ilmσ  † (ˆs ilm = αγ aiLα (σ /2)αγ aiLγ ), σ being the Pauli spin matrices. As shown earlier in Sect. 2.3.3, we can derive the first-principles tight-binding (TB) Hamiltonian on the basis of the LDA to the DFT and the tight-binding linear

106

3 Metallic Magnetism at Finite Temperatures

muffin-tin orbital (TB-LMTO) method. The Hamiltonian is written as (see (2.167)) 

HLDA =

iLj L σ

† HiLj L aiLσ aj L σ ,

      1 HiLj L = χiL  − ∇ 2 + v(r) χj L = εiL δij δLL + tiLj L . 2

(3.231)

(3.232)

Here χiL ’s are the nearly orthogonal basis functions for orbital L on site i. v(r) is a LDA potential, εiL is an atomic level, and tiLj L is a transfer integral between orbitals χiL and χj L . When we construct the tight-binding parameters for noninteracting system from the TB-LMTO LDA Hamiltonian, we have to take into account the fact that the oneelectron Hamiltonian (3.232), especially the atomic level εiL , contains the effects of strong intraatomic Coulomb interactions. We therefore subtract the contribution of electron–electron interactions from the LDA Hamiltonian (3.232) via the relation U . Here E U H0  = ELDA − ELDA LDA is the ground-state energy in the LDA, and ELDA 0 is a LDA functional to the intra-atomic Coulomb interactions. The atomic level εiL for the noninteracting system is then obtained from the relation [63], 0 = εiL

U ∂ELDA ∂ELDA − . ∂niLσ ∂niLσ

(3.233)

Here niLσ is the charge density at the ground state. For the transfer integral, we 0  adopt the approximation tiLj L = tiLj L expecting small corrections. U Several forms of ELDA have been proposed. Among them, we adopt here the Hartree–Fock type form [70], since we are considering an itinerant electron system where the ratio of the Coulomb interaction to the d band width is not larger than one. U ELDA =

1  1    U nj d nj d + (U − J )nj d nj d . 2 2   σ j mm σ

(3.234)

j mm

 Here nj d = mσ nj lmσ /2(2l + 1) with l = 2. U and J are the orbital-averaged Coulomb and exchange interactions defined by U=

 1 Umm , 2 (2l + 1) 

(3.235)

mm

(U − J ) =

 1 (Umm − Jmm ), 2l(2l + 1) 

(3.236)

mm

where Umm and Jmm are orbital dependent intra-atomic Coulomb and exchange 0 integrals for d electrons. From (3.233) and (3.234), we obtain the atomic level εiL

3.7 First-Principles Dynamical CPA and Metallic Magnetism

for the noninteracting system as     1 1 1 0 1− U− J nd δl2 . εiL = εiL − 1 − 2(2l + 1) 2 (2l + 1)

107

(3.237)

Note that nd denotes the total d electron number per atom. The Coulomb and exchange energy parameters, U and J , may be obtained by the following steps. We introduce an orbital dependent energy functional ELDA+U as follows. U ELDA+U = ELDA + EU − ELDA .

(3.238)

U Here ELDA is the LDA total energy, ELDA is the Hartree–Fock type LDA contribution given by (3.234), and EU is the orbital-dependent contribution due to d electrons, which is given by

EU =

1  1    U nj dmσ nj dm σ  + (U − J )nj dmσ nj dm σ . (3.239) 2 2   σ j mm σ

j mm

Note that ELDA+U reduces to the original LDA energy when nj dmσ = nj d . The variational principle δELDA+U − μδN = 0 yields the Kohn–Sham potential as  U (nj dm −σ − nj ) vdmσ (r) = vLDA (r) − μ + +



m

(U − J )(nj dm σ − nj ).

(3.240)

m

Here vLDA (r) is the LDA potential given by (2.116). A method to take into account the orbital dependence of the LDA potential with use of the intra-atomic Coulomb interactions is called the LDA + U method [63]. The above potential suggests that the following atomic levels in the TB-LMTO (see (2.167)) are orbital dependent,       1 (3.241) εj Lσ = χj Lσ  − ∇ 2 + vLσ (r) χj Lσ , 2 and ∂εj L↑ /∂nj L ↓ = ∂εj L↓ /∂nj L ↑ = U , ∂εj L↑ /∂nj L ↑ = ∂εj L↓ /∂nj L ↓ = (U − J )(1 − δmm ) for l = 2. For the evaluation of U and J , we remove the charge transfer between the d orbitals and the other orbitals, and change the d electron number nj dmσ or the spin density mj dm = nj m↑ − nj m↓ as δnj dmσ  or δmj dm . Then we can calculate the selfconsistent change of the LDA atomic level as {δεj dmσ }. The U and J , which are screened by the other electrons, may be obtained as U=

δεj dm↑ , δnj dm ↓

(3.242)

108

3 Metallic Magnetism at Finite Temperatures

J=

δ(εj dm↑ − εj dm↓ ) . δmj dm

(3.243)

The method mentioned above to obtain U and J is known as the constraint LDA [70]. The intraatomic Coulomb and exchange interaction energy parameters U0 , U1 , and J in the Hamiltonian (3.228) are obtained from U and J in the LDA+U via the relations: U0 = U + 8J /5, U1 = U − 2J /5 and J = J , where we adopted the relation U0 = U1 + 2J obtained from the rotational invariance. We can apply the dynamical CPA to the first-principles model Hamiltonian (3.228). The Hubbard–Stratonovich transformation (3.24) is then extended as follows.

    det A a A a − mm (xm Amm xm −2am Amm xm )    e mm m mm m = dx . (3.244) m e πM m Here {aμ } are the Bose-type operators which commute each other. Amm is a M × M matrix, and {xm } are auxiliary field variables. Discretizing the time in the free energy in the interaction representation (see (3.31)) and applying the Hubbard–Stratonovich transformation (3.244) to the Bose-type operators at each time under the T -product, one can obtain the functional integral form of the free energy to the Hamiltonian (3.228) as follows.

e

−β F

=

  N 2l+1 





δξ im (τ )δζim (τ ) Z 0 ξ (τ ), ζ (τ )

i=1 m=1



1   × exp − 4  i mm



β

 dτ ζim (τ )Aimm ζm (τ )

0

+

xyz 

 α ξimα (τ )Bimm  ξim α (τ )

.

(3.245)

α

Here ζim (τ ) and ξ im (τ ) denote the time-dependent charge and exchange fields acting on site i and orbital m. Z 0 (ξ (τ ), ζ (τ )) is a partition function given by (3.35) in which the Hamiltonian H (τ, ξ(τ ), η(τ )) has been replaced by  

 1 0 εiL − μ − H τ, ξ (τ ), −iζ (τ ) = iAimm ζim (τ )δl2 nˆ iL (τ ) 2  iL m   1  α α α  δ − Bimm ξ (τ ) + h m ˆ (τ )  im α l2 iL im 2  α m  † tiLj L aiLσ (τ )aj L σ (τ ). (3.246) + iLj L σ

3.7 First-Principles Dynamical CPA and Metallic Magnetism

109

α Here Aimm and Bimm  (α = x, y, z) are the Coulomb and exchange energy matrix elements defined by

Aimm = U0 δmm + (2U1 − J )(1 − δmm ),

(3.247)

α (α = x, y), Bimm  = J (1 − δmm ) z Bimm  = U0 δmm + J (1 − δmm ).

(3.248) (3.249)

We express in the next step the free energy with use of the Matsubara frequencies, introduce the coherent potential (Σ)iLnσj L n σ  = ΣLσ (iωn )δij δLL δnn δσ σ  , and make the single-site approximation in order to reduce the number of variables. Neglecting the out-of-phase thermal spin fluctuations between different orbitals on a site, we obtain the free energy per site as follows.    β J˜α −1 ˜ FCPA = F − β ln (3.250) dξα e−βEeff (ξ ) . 4π α  Here ξα ≡ m ξimα is the static exchange field. The effective exchange energy parameters are defined by J˜x = J˜y = J˜⊥ = [1 − 1/(2l + 1)]J , J˜z = U0 /(2l + 1) + J˜⊥ . The effective potential consists of the static part and the dynamical part; Eeff (ξ ) = Est (ξ ) + Edyn (ξ ). The former is given by 



1 ln 1 − δvL↑ (0)FL↑ (iωn ) 1 − δvL↓ (0)FL↓ (iωn ) β mn 1 − J˜⊥2 ξ⊥2 FL↑ (iωn )FL↓ (iωn ) 4   1 2 2 2 2 ˜ ˜ + −(U0 − 2U1 + J ) n˜ L (ξ ) − (2U1 − J )n˜ l (ξ ) + J⊥ ξ⊥ + Jz ξz . 4 m

Est (ξ ) = −

(3.251) Here δvLσ (0) = vLσ (0) − ΣLσ (iωn ), and vLσ (0) is the static potential for electrons with orbital L and spin σ . FLσ (iωn ) denotes the coherent Green function for orbitalL and spin σ . The charge  densities, n˜ L (ξ ) and n˜ l (ξ ) are defined by n˜ L (ξ ) = σ n˜ Lσ (ξ ) and n˜ l (ξ ) = m n˜ L (ξ ), respectively. Note that the spin-flip contributions as well as the inter-orbital contributions characterized by the parameters 2U1 − J and J˜⊥ now appear in the energy expression. Equation (3.251) is an extension of the static potential (3.149) to the degenerate case. The dynamical contribution to the effective potential is given by e−βEdyn (ξ ) =D≡

  ∞ % 2l+1 β det B α  n=1

α

(2π)2l+1

m

& 2l+1 det A  2 β d ξmα (iωn ) d ζm (iωn ) (2π)2l+1 m 2

110

3 Metallic Magnetism at Finite Temperatures

  β  ∗ × D exp − ζm (iωn )Amm ζm (iωn ) 4  n =0 mm

+



 ∗ α ξmα (iωn )Bmm ,  ξm α (iωn )

(3.252)

α

and    v˜Lσ σ  (iωn − iωn )g˜ Lσ  L σ  (iωn ) . (3.253) D = det δnn δLL δσ σ  − σ 

Here ζm (iωn ) (ξmα (iωn )) denotes the dynamical charge (exchange) field for orα bital m. The matrices Amm and Bmm  (α = x, y, z) have been defined by (3.247), (3.248), and (3.249). v˜Lσ σ  (iωn −iωn ) is the dynamical potential, and g˜ Lσ L σ  (iωn ) is the Green function in the static approximation. The dynamical CPA equation is obtained from the stationary condition of the free energy δFCPA /δΣiLσ (iωn ) = 0 as follows. (i)

GLσ (iωn ) = FLσ (iωn ), 

" (i)

GLσ (iωn ) = g˜ Lσ Lσ (iωn ) +

ν

δD ν # κLσ (iωn )δΣLσ (iωn ) .  1+ (D ν − 1) eff

(3.254)

(3.255)

ν (i)

Here GLσ (iωn ) is the Green function with an impurity dynamical potential embedded in the effective medium. Dν is the determinant for the scattering matrix due to dynamical potential v˜Lσ σ  (±iων ). κLσ (iωn ) = 1 − FLσ (iωn )−2 HLσ (iωn ) and HLσ (iωn ) = δFLσ (iωn )/δΣLσ (iωn ). The average ∼eff at the r.h.s. of (3.255) denotes a classical average with respect to the effective potential Eeff (ξ ). The local charge and magnetic moment are derived from the free energy (3.250) as follows. nˆ L  =

1 FLσ (iωn ), β nσ

(3.256)

m ˆ zL  =

1 σ FLσ (iωn ). β nσ

(3.257)

In particular, the l = 2 components of local charge and magnetic moment (i.e.,   ˆ l  = m m ˆ L ) are expressed as nˆ l  = m nˆ L  and m nˆ l  = n˜ l (ξ )eff ,

(3.258)

ˆ l  = ξ eff . m

(3.259)

3.7 First-Principles Dynamical CPA and Metallic Magnetism

111

Fig. 3.11 Effective potential for bcc Fe at the temperature T /TC = 0.5 on the ξx –ξz plane [72]

The last expression of magnetic moment indicates a physical picture as shown in Fig. 3.1 that both the longitudinal and transverse components of local moments fluctuate in the metallic system with elevating temperature. Typical Coulomb and exchange interactions calculated by the LDA + U constraint method are reported to be U = 0.17 Ry and J = 0.066 Ry for bcc Fe, and U = 0.22 Ry and J = 0.066 Ry for fcc Ni [73]. These values considerably depend on the method of calculations. Recent calculations based on the random phase approximation report that the exchange interaction energy parameters can be reduced by 30 % irrespective of transition metal elements [74]. According to the full Hartree–Fock calculations for solids, on the other hand, the averaged bare Coulomb and exchange interactions are 1.55 Ry and 0.060 Ry (1.66 Ry and 0.065 Ry) for bcc Fe (fcc Ni) [75]. The results imply that the LDA + U Coulomb interaction energy parameters are screened by 4s–4p electrons by a factor of five or ten, while the screening on the exchange interactions is negligible. Figure 3.11 shows the effective potential of ferromagnetic Fe calculated by the first-principles dynamical CPA. We adopted here the interaction energy parameters, U = 0.17 Ry and J = 0.066 Ry. Dynamical corrections are taken into account up to the 4-th orders in Coulomb interaction strength in the calculations. The dynamical potential Edyn (ξ ) acts to reduce the longitudinal amplitude of magnetic moments and enhance the transverse spin fluctuations. The calculated potential for ferromagnetic Fe has a double minimum structure. This implies that the local magnetic moments of Fe show large thermal spin fluctuations which change the magnetic moments in direction. The behavior differs from the case of the single-band model where quantum spin fluctuations are more significant so that the potential shows a single minimum structure (see Fig. 3.7). The magnetization vs. temperature curves of Fe are presented in Fig. 3.12. Obtained ground-state magnetization 2.58 μB is considerably larger than the experimental value 2.22 μB . Calculated Curie temperatures are summarized in Table 3.2. The Curie temperature in the Hartree–Fock approximation is 12200 K. The static approximation reduces TC by a factor of 6. Dynamical corrections in the HA further

112

3 Metallic Magnetism at Finite Temperatures

Fig. 3.12 Magnetization vs. temperature curves (M − T ), inverse susceptibilities (χ −1 ), and the amplitudes of local magnetic moments (m2 1/2 ) for bcc Fe in the dynamical CPA (solid curves) [72]. The magnetizations calculated by the DMFT without transverse spin fluctuations are shown by open squares [73]. Experimental M − T curve is shown by + points [76]

reduce TC and yield TC = 1930 K [72]. The DMFT calculations without transverse spin fluctuations also yield approximately the same value T = 1900 K [73]. These values are still higher than the experimental value (1040 K) by a factor of 1.8. The first-principles dynamical CPA also overestimates the Curie temperature of fcc Co by a factor of 1.8 as found on Table 3.2. The inverse susceptibility above TC follows the Curie–Weiss law. The dynamical CPA yields the effective Bohr magneton number 3.0 μB , which agrees with the experimental value 3.2 μB [77]. The amplitudes of local magnetic moments m2 1/2 show a weak temperature dependence and take a value 3.1 μB at 2000 K; the calculated effective Bohr magneton number approximately agrees with the amplitude of local moment, so that the Rhodes–Wohlfarth ratio (meff /m2 1/2 ) is 1 in agreement with the experimental fact. An alternative example of ferromagnetic metals is the fcc Ni. The effective potential for Ni shows a single minimum in both the ferro- and para- magnetic states as shown in Fig. 3.13. It indicates small thermal spin fluctuations around the equilibrium point. The dynamical contribution to the effective potential acts as an effective magnetic field which weakens the spin polarization. Calculated ground-state magnetization 0.63 μB is close to the experimental value 0.62 μB . Here the Coulomb and exchange energy parameters are chosen to be U = 0.22 Ry and J = 0.066 Ry. The magnetization vs. temperature curve is presented in Fig. 3.14. We find the Curie temperature TC = 620 K. As summarized in Table 3.2, calculated Curie temperatures of Ni in the Hartree–Fock approximation is 4940 K, the static approximation reduces TC by a factor of 3 or 4 (1420 K), and the first-principles dynamical CPA with use

3.7 First-Principles Dynamical CPA and Metallic Magnetism

113

Fig. 3.13 Effective potential of Ni at T /TC = 0.8 [72]

of the HA yields TC = 620 K [72], being in good agreement with the experimental value 630 K. The calculated inverse susceptibility for Ni follows the Curie–Weiss law, and shows an upward convexity in the high-temperature region, being in agreement with the experimental data. The effective Bohr magneton number calculated at T ∼ 2000 K is 1.6 μB in the dynamical CPA. The result is in good agreement with the experimental value 1.6 μB [81]. Calculated amplitude of local moment m2 1/2 slightly increases with increasing temperature and takes a value 1.97 μB at 1000 K, which is larger than 1.27 μB , the value in the local moment model. Note that the d electron number in the metallic state is nd = 8.7 because of the hy-

Fig. 3.14 Magnetization vs. temperature curves (M − T ), inverse susceptibilities (χ −1 ), and the amplitude of local magnetic moments (m2 1/2 ) for Ni in the dynamical CPA [72]. Experimental data of magnetization curve are shown by + [82]. The magnetizations calculated by the DMFT without transverse spin fluctuations are shown by open squares [73]

114

3 Metallic Magnetism at Finite Temperatures

Fig. 3.15 Effective Bohr magneton numbers in various 3d transition metal alloys as a function of conduction electron number per atom. Large closed circles show the theoretical values of Fe, Co, and Ni obtained by the dynamical CPA [72] Table 3.2 Curie temperatures TC for Fe, Co, and Ni calculated by the Hartree–Fock approximation (HF), the dynamical CPA with static approximation (SA), and the dynamical CPA with the harmonic approximation (HA) [72]. The experimental data (Expt.) are shown on the bottom line [78–80] TC (K)

Fe

Co

Ni

HF

12200

12100

4940

SA

2070

3160

1420

HA

1930

2550

620

Expt.

1040

1388

630

bridization of the d bands with the sp bands. It is smaller than nd = 9.4 expected from a d-band model with strong ferromagnetism. The amplitude of the local magnetic moment is therefore larger than the value expected from the local moment √ model, 1.27μB (= M(0)(M(0) + 2), where M(0) is the ground-state magnetization 0.62 μB ). Although the quantitative agreement of TC is not obtained in the single-site theory, the paramagnetic susceptibilities at high temperatures are quantitatively described by the first-principles dynamical CPA. Experimental data of effective Bohr magneton numbers in 3d transition metal alloys continuously change with the conduction electron number per atom as shown in Fig. 3.15. Calculated effective Bohr magneton numbers, 3.0 μB (Fe), 3.0 μB (Co), and 1.6 μB (Ni) are in the experimental data.

Chapter 4

Magnetic Excitations

The ferromagnetic ground state is excited by applying a time-dependent magnetic field or elevating temperature. These excited states are observed by various experimental methods such as neutron scattering and nuclear magnetic resonance techniques. Here we consider the low-energy excitations from the ferromagnetic ground state in both metals and insulators. We introduce first the spin wave excitations in the local moment system which are instructive for understanding the nature of the excitations. Next, we will show that itinerant electron ferromagnets cause the same type of collective excitations, as well as the spin-flip excitations of each electron called the Stoner excitations. Finally, we introduce the dynamical susceptibility in order to generalize the theory of magnetic excitations, and treat the same topics with use of the susceptibility in the random phase approximation (RPA).

4.1 Spin Waves in the Local Moment System We consider in this section the local moment system described by the Heisenberg model, and clarify the basic concept of spin waves as low-energy excitations. In the atomic system with n electrons in the unfilled shell, the atomic moment is built up according to the Hund rule (see Sect. 1.3). It consists of the spin magnetic moment with spin S and its z component M, as well as the orbital moment with the angular momentum L and its z component ML . In the crystalline system in which the spin-orbit interaction is negligible, the crystal field removes the orbital degeneracy, so that the orbital moments are quenched (see Sect. 1.5). We may then express the magnetic states of the insulator by means of a set of {Mi }, where Mi denotes the z component of the spin of magnetic ion i. The atomic spin on site i is given by Si =

1  † aimσ (σ )σ σ  aimσ  , 2 

(4.1)

mσ σ

Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_4, © Springer-Verlag Berlin Heidelberg 2012

115

116

4 Magnetic Excitations

† where aimσ (aimσ ) denotes the creation (annihilation) operator for an electron on orbital m in the unfilled shell. The eigen states |SMi  of S 2i and Siz are defined by S i |SMi  = S(S + 1)|SMi  and Siz |SMi  = Mi |SMi . Mi takes 2S + 1 values from −S to S. We assume that the ferromagnetic state of the insulator is expressed by a Heisenberg model as follows.

H = −2J

NN  (i,j )

S i · S i = −2J

NN  

1 2 Siz + Si+ Sj− + Si− Sj+ . 2

(4.2)

(i,j )

Here the sum is taken with respect to the nearest-neighbor (NN) pairs for simplicity. Si± (= Six ± iSiy ) are the raising and lowering operators of spin on atom i, satisfying √ the relation Si± |SMi  = S(S + 1) − Mi (Mi ± 1)|S Mi ± 1. Note that J > 0 since we assume the ferromagnetic ground state.  As in the case of the Hund-rule coupling, the magnitude of total spin S tot = i S i should be maximized at the ground state: Stot = N S. Here N is the number of magnetic ions. The ground state wave function is therefore given by   Ψ0 = {Mi = S} . (4.3) The state Ψ0 is in fact verified to be the ground state, and its energy is given by E0 = −N J zS 2 where z is the number of the nearest neighbors. H Ψ0 = E 0 Ψ0 .

(4.4)

Note that the ground-state energy E0 agrees with what is expected from the classical Heisenberg model. The low-energy excitations from the ground state may be created by the one-spin flipped states defined by 1 |i = √ Si− Ψ0 = |S, . . . , S, Mi = S − 1, S, . . .. 2S

(4.5)

Note that these states are orthogonal to the ground state; i|Ψ0  = 0, and are also orthogonal to each other; i|j  = δij . The Hamiltonian is diagonalized in the subspace {|i}. In fact, we can verify that NN of i 

H |i = −N J zS 2 + 2J zS |i − 2J S |l.

(4.6)

l

Therefore we have i|H |j  = (E0 + 2J zS)δij − 2Jij S.

(4.7)

Here we have defined Jij as Jij = J for the NN pair (i, j ), and otherwise Jij = 0.

4.1 Spin Waves in the Local Moment System

117

Since (4.7) is the same type of Hamiltonian matrix as the one-electron tightbinding model (1.53), we can diagonalize it with use of the Bloch wave function |k as follows. 1  −ik·Rj e |j , |k = √ N j

(4.8)

H |k = (E0 + ωk ) |k.

(4.9)

Here ωk is the eigen value for excited states,

ωk = 2J zS 1 − γ (k) ,

(4.10)

and

 and γ (k) = z−1 NN j exp(ik · R j ). The excitation energy ωk is known as the spin wave energy because the dynamics associated with the excitations form a wave of transverse spin components. Note that ωk → 0 as k approaches 0, so that the excitation energy approaches the ground state energy. The eigen state |k is obtained from the ground state Ψ0 by applying the spin-flip operator Sk− as follows. 1 |k = √ Sk− Ψ0 , 2S

(4.11)

1  −ik·Rj − Sk− = √ e Sj . N j

(4.12)

where

Let us verify that the dynamics associated with low-energy excitation are spin waves. In the Heisenberg representation, the spin dynamics is determined by the equation of motion as follows. i

 dSk−  − = Sk , H . dt

(4.13)

For a low energy state Ψ0 such as the ground state, we have [Sk− , H ]Ψ0 = −ωk Sk− Ψ0 . We have then i

dSk− ≈ −ωk Sk− . dt

(4.14)

Thus, 1  − Sk− (t) = Sk− (0) eiωk t = √ Sj (0) e−i(k·R j −ωk t) . N j

(4.15)

118

4 Magnetic Excitations

Fig. 4.1 Spin-wave motion associated with the low-energy excitations in the local moment system

In the real space, this means that 1  − Sj− (t) = √ Sk (0) ei(k·R j +ωk t) . N k

(4.16)

Since Sj− = Sj x − iSjy , we obtain 1  − Sj x (t) = √ Sk (0) cos(k · R j + ωk t), N k 1  − Sk (0) sin(k · R j + ωk t). Sjy (t) = − √ N k

(4.17)

Thus the motion of spins associated with low-energy excitations in the ferromagnets behaves as a wave of transverse spins (see Fig. 4.1).

4.2 Spin Waves in Itinerant Ferromagnets The spin wave excitations are also possible in the itinerant ferromagnets. We examine in this section the low-energy magnetic excitations in itinerant electron systems. We adopt the Hubbard model (1.51) in the following:    † H= ε0 niσ + tij aiσ aj σ + U ni↑ ni↓ . (4.18) iσ

ij σ

i

We assume that the ground state is ferromagnetic, and that it is approximately described by the Hartree–Fock wave function Ψ0 , for simplicity.  occ  occ   †  † Ψ0 = ak↑ ak↓ |0. (4.19) k

k

The Hartree–Fock ground state Ψ0 and the ground-state energy E0 then approximately satisfy the following equation as previously discussed in Sect. 2.1. H Ψ0 ≈ E 0 Ψ0 .

(4.20)

4.2 Spin Waves in Itinerant Ferromagnets

119

Here E0 =

occ  kσ

1 εkσ − N U n20 − m20 . 4

(4.21)

Here εkσ = ε0 + U n0 /2 − U m0 σ/2 + εk is the Hatree–Fock one-electron energy, εk being the eigenvalue for the transfer integral matrix tij . N denotes the number of lattice points. As in the local moment case, we may introduce the spin-flip operator Si− = Six − † iSiy = ai↓ ai↑ . In the momentum representation, it is written as 1  iq·R i − 1  iq·R i  − Si− = √ e Sq = e Sqk . N q N q k Here Sq− is given by Sq− =



(4.22)

√ − N , and Sqk is the spin-flip operator defined by

− k Sqk /

† − Sqk = ak+q↓ ak↑ .

(4.23)

Since Sq− Ψ0 described a spin-wave excited state in the case of the insulator model (see (4.11)), we assume here that the low energy excitations in the itinerant system − Ψ0 } as follows. are described by a superposition of the spin-flip states {Sqk Ψq =



− ck Sqk Ψ0 .

(4.24)

k

The eigen value equation is written as H Ψq = (E0 + ωq )Ψq .

(4.25)

Here {ck } and ωk are coefficients and excitation energy to be determined. Note that − − Ψ0  = 0 and Sq− k  Ψ0 |Sqk Ψ0  = nk↑ (1 − nk+q↓ )δqq  δkk  . Here nkσ is the Ψ0 |Sqk electron occupation number for an electron with momentum k and spin σ . In order to examine the excited states, we express the Hamiltonian in the momentum representation as follows. H = H0 + HI ,  εk0 nkσ , H0 =

(4.26) (4.27)

kσ

HI =

U 2N

 k1 k2 qσ σ 

ak†1 +qσ ak†2 −qσ  ak2 σ  ak1 σ .

(4.28)

Here εk0 = ε0 + εk is the one-electron energy for the noninteracting Hamiltonian.

120

4 Magnetic Excitations

In the calculation of H Ψq at the l.h.s. of (4.25), we have  − − −

H Sqk + Sqk Ψ0 = H, Sqk H Ψ0 .

(4.29)

In the second term at the r.h.s. of the above equation, we may adopt (4.20); − − − Sqk H Ψ0 ≈ E0 Sqk Ψ0 . The commutation relation [H, Sqk ] in the first term at the r.h.s. is obtained as follows by using (4.27) for H0 and (4.28) for HI , respectively.  0

− − H0 , Sqk = εk+q − εk0 Sqk , (4.30)

 U  † † − = ak+q+q  ↓ ak† −q  σ  ak  σ  ak↑ − ak+q↓ ak† −q  σ  ak  σ  ak−q  ↑ . HI , Sqk N    kqσ

(4.31) − The interaction part [HI , Sqk ] expands the Hilbert space when it is applied to the ground state Ψ0 . We take the diagonal part among the k  scattering terms, and neglect the other terms that expand the space. This is called the Random Phase Approximation (RPA). For example,    †  † † †  ak+q+q  ↓ ak  −q  ↑ ak ↑ ak↑ Ψ0 ≈ ak+q+q  ↓ ak↑ ak+q  ↑ ak↑ Ψ0 q

k

q

= −nk↑

 k

− Sqk 

 Ψ0 ,

(4.32)

taking only one term k  = k + q  among various k  terms, which is conjugate with ak↑ . Here nkσ is the electron occupation number at the ground state. Making use of the RPA, we obtain

− 0 − H Sqk Ψ0 = εk+q − εk0 Sqk Ψ0 +

U − (nk  ↑ − nk  +q↓ )Sqk Ψ0 N  k

−

 U − − (nk↑ − nk+q↓ ) Sqk  Ψ0 + E0 Sqk Ψ0 . N 

(4.33)

k

Substituting the wave function (4.24) into the eigen value equation (4.25) and applying the above relation (4.33) as well as the orthogonality relation − − Sqk  Ψ0 |Sqk Ψ0  = nk↑ (1 − nk+q↓ )δkk  , we obtain the eigen value equation for {ck } as follows. 0

U εk+q − εk0 + U m − ωq ck = (nk  ↑ − nk  +q↓ )ck  . (4.34) N  k

4.2 Spin Waves in Itinerant Ferromagnets

121

Fig. 4.2 Stoner excitations ωqk = εk+q↓ − εk↑ in itinerant electron system

(1) Stoner excitations Equation (4.34) has the simple solutions ck = 0 and ck  = 0 for the other k  , which correspond to individual electron excitations. The wave function is given by † − Ψ0 = ak+q↓ ak↑ Ψ0 . Ψqk = Sqk

(4.35)

The excitation energy as the eigen value is given by 0 − εk0 + U m = εk+q↓ − εk↑ , ωqk = εk+q

(4.36)

where εkσ is the Hartree–Fock one-electron energy for σ spin electron. This is an individual excitation which excites one electron from k ↑ to k + q ↓, and is known as the Stoner excitation (see Fig. 4.2). Stoner excitations ωqk yield the exchange splitting ωq=0 ≡ Δ = U m at q = 0, and form a band for a given q = 0. In the free-electron model band, we have



  −2kF q + q 2 + Δ ≤ ωqk ≤ 2kF q + q 2 + Δ. 2m 2m

(4.37)

The region of the Stoner excitations as a function of q is schematically shown in Fig. 4.3 by hatched lines. (2) Spin wave excitations There are alternative solutions of low energy excitations with {ck = 0} in the eigen value equation (4.34). After having divided the eigenvalue equation (4.34) 0 − εk0 + U m − ωq ), we multiply nk↑ − nk+q↓ and sum up the equation by (εk+q with respect to k. We then find the equation for excitation energy ωq as follows. 1=

nk↑ − nk+q↓ U  . 0 N ε − εk0 + U m − ωq k+q k

(4.38)

There are many solutions in the case of the strong ferromagnet (large U and 0 − εk0 + U m > 0 and we have each solution between neighnk↑ = 1), because εk+q boring zero points of the denominator. These solutions correspond to the Stoner

122

4 Magnetic Excitations

Fig. 4.3 Spin wave excitations (thick line which starts from ωq=0 = 0) and Stoner excitations (hatched region) as a function of the wave vector q in itinerant electron system

excitations. However, we have one more solution below the Stoner continuum solu0 tions as shown in Fig. 4.3. This solution ωq satisfies εk+q − εk0 + U m > ωq , and 0 ωq −→ 0 as q goes to zero, as verified from (4.38). For small q, εk+q − εk0 − ωq is small as compared with Δ(= U m). Thus expanding the r.h.s. of (4.38) with 0 respect to εk+q − εk0 − ωq , we obtain the solution for small q as ωq =

0 1  (nk↑ − nk+q↓ ) εk+q − εk0 . N m

(4.39)

k

For the system with cubic symmetry, we obtain ωq = Dq 2 , D=

(4.40)

∂ 2 εk 1  (nk↑ + nk↓ ) 2 . 2M ∂kx

(4.41)

k

This is the same dispersion as the one found in the spin wave in the Heisenberg model. The coefficient D is called the spin wave stiffness constant. The wave function for ωq = Dq 2 is obtained by substituting the eigenvalue into (0) (1) (2) (4.34) and expanding the coefficients ck as ck = ck + ck + ck + · · · according (0) to the magnitude of q. Up to the first order, we have ck = 0 and ck(1) = const. Thus we obtain Ψq = Sq− Ψ0 .

(4.42)

This is the same form as the wave function of the spin wave in the Heisenberg model (see (4.11)). In the spin-wave excited state Ψq , total magnetization is reduced by two in unit of the effective Bohr magneton number as compared with that in the ground state. Taking the same steps as in the spin wave of the Heisenberg model, we can verify that the motions of transverse spins associated with such low energy excitations are

4.2 Spin Waves in Itinerant Ferromagnets

123

given as follows (see (4.17)). 1  − Sq (0) cos(q · R j + ωq t), Sj x (t) = √ N |q|
1 Sjy (t) = − √ N



(4.43)

Sq− (0) sin(q · R j + ωq t).

|q|
Here we introduced a cut-off frequency qc , below which the spin wave excitation energy ωq = Dq 2 is obtained. In conclusion, the spin-wave excitations exist in the long-wave limit even in the itinerant-electron ferromagnet. They correspond to the corrective motion given by (4.43). It should be noted that the Stoner excitations are included in the eigen states of the Hartree–Fock Hamiltonian (2.7). Associated temperature dependence of the magnetization is obtained from the self-consistent equations (2.11) and (2.12), or (3.1) and (3.2). At low temperatures, one can obtain the temperature dependence of the magnetization per atom as follows solving the self-consistent equations. m(T ) = m(0) +

(1) −1 (1) −1 π 2 ρ↑ ρ↑ − ρ↓ ρ↓ 2 T + ··· . 3 ρ↑−1 + ρ↓−1 − U

(4.44)

Here ρσ (ρσ(1) ) denotes the density of states (DOS) at the Fermi level in the ferromagnetic ground state (the first derivative of the DOS at the Fermi level). The above expression indicates that the Stoner excitations yield the temperature variations being proportional to T 2 . The same conclusion is also verified from (3.22) for the weak ferromagnet. In the spin wave excitations in the strong ferromagnets, we have the eigen-value equation (4.25) with the energy ωq = Dq 2 and the excited state Ψq = Sq− Ψ0 for small |q| < qc . The equation is written as [H, Sq− ]Ψ0 = ωq Sq− Ψ0 . It implies that the following commutation relations are satisfied for low-energy excited states in the strong ferromagnets.   (4.45) H, Sq− ≈ ωq Sq− . Then we can find the higher excited states Ψnq = (Sq− )n Ψ0 such that H Ψnq = (E0 + nωq )Ψnq . These are the higher-order excited states of spin waves called magnon excitations, and the associated magnetization is reduced as Mz Ψnq = (M0 − 2n)Ψnq . Here M0 denotes the ground-state magnetization. We can generalize these results as   |q|


nq ωq Ψ {nq } , (4.46) H Ψ {nq } = E0 + q



 |q|


Mz Ψ {nq } = M0 − 2 nq Ψ {nq } . q

(4.47)

124

4 Magnetic Excitations

Here the spin-wave excited states are given by |q|
Ψ {nq } = Sq− q Ψ0 .

(4.48)

q

Associated temperature dependence of the magnetization is calculated from (4.47) as Mz  = M0 − 2

|q|
nq .

(4.49)

q

Here the thermal average of the magnon number nq  is obtained as nq  = 1/(eβωq − 1). Thus we find the magnetization per atom m(T ) = Mz /L as follows.    3 T 3/2 Ω m(T ) = m(0) − ζ + ··· . (4.50) 2 2 πD Here Ω is the volume of the unit cell and we used the relation ωq = Dq 2 . ζ (3/2) is Riemann’s ζ function defined by √     ∞ ∞ π 1 x2 3 = . (4.51) dx x =ζ 3/2 e − 1 2 2 n 0 n=1

Note that the spin wave excitations are not included in the single-site theory presented in Chap. 3, but govern the temperature dependence of the magnetization in the low temperature limit since the spin wave excitations yield the T 3/2 law while the Stoner excitations yield the T 2 law, as seen from (4.50) and (4.44).

4.3 Dynamical Susceptibility Magnetic excitations are described by the dynamical susceptibility. This implies that one can describe the excitations discussed in the last section by means of the susceptibility. In this section we introduce dynamical susceptibility on the basis of the linear response theory and clarify its properties. Let us assume that the system is perturbed by a time-dependent force F (t), whose interaction is given by H1 (t) = −AF (t).

(4.52)

Here A is an operator of a physical quantity. According to the linear response theory [83], the linear change of the physical quantity B(t) due to the external timedependent perturbation H1 (t) is expressed as follows (see Appendix E).  t



B(t) = χBA t − t  F t  dt  . (4.53) −∞

4.3 Dynamical Susceptibility

125

The linear response function χBA (t) is given by a time correlation function at equilibrium state as follows. i χBA (t) = [BH (t), A]. 

(4.54)

Here BH (t) is the Heisenberg representation of the physical quantity B, i.e., BH (t) = exp(iH t/) B exp(−iH t/). We consider here the response of the magnetic moments mi on site i under the time-dependent magnetic fields {hj (t)}. The interaction (4.52) is then given by  H1 (t) = − mj · hj (t). (4.55) j

The response (4.53) of the magnetic moment on site i is given by  t



αγ miα (t) = χij t − t  hj γ t  dt  . jγ

−∞

(4.56)

αγ

The response function χij (t) (α, γ = x, y, z) is given by i αγ χij (t) = [mHiα (t), mj γ ]. 

(4.57)

When we apply the magnetic field oscillating with a frequency ω as hj (t) = hj e−i(ω+is)t , we have the following response from (4.56).  αγ miα (t) = χij (ω + is)hj γ e−i(ω+is) t .

(4.58)

(4.59)

jγ

Here s is the positive-definite infinitesimal number. It implies that we apply the infinitesimally small field at t = −∞, and increase it continuously up to t = 0. The αγ response function χij (ω + is) is given as αγ χij (ω + is) =

 0

∞

αγ

χij (t) ei(ω+is)t dt.

(4.60)

The magnetic field and the magnetic moments are expressed by the Fourier lattice series as follows.  hj = h(q, ω) eiq·Rj , (4.61) q

mi (t) =

 q

m(q, ω) ei(q·Ri −ω t) .

(4.62)

126

4 Magnetic Excitations

Substituting the above expressions into (4.59), we obtain the linear response in Fourier representation as follows. mα (q, ω) =



χ αγ (q, ω + is)hγ (q, ω).

(4.63)

γ

Here we assumed the translational symmetry of the system in the equilibrium state. The response function χ αγ (q, ω + is) is defined as χ αγ (q, ω + is) =



χij (ω + is)e−iq·(Ri −R j ) . αγ

(4.64)

i

Substituting (4.60) with (4.57) into (4.64), we obtain   i ∞  αγ χ (q, ω + is) = L mHα (q, t), mγ (−q) ei(ω+is) dt.  0

(4.65)

Here L is the number of lattice points. The operator m(q) is the Fourier component of the local magnetic moment mi . mα (q) =

1 miα e−iq·R i . L

(4.66)

i

The response function χ αγ (q, ω + is) is called the dynamical susceptibility and determines the linear response to the time-dependent perturbation H1 (t) = −L



mγ (−q) hγ (q, ω) e−i(ω+is)t .

(4.67)

γ

Here hγ (q, ω) is the Fourier transform of the magnetic field hi . hγ (q, ω) =

1 hiγ e−iq·R i . L

(4.68)

i

It should be noted that the interaction H1 (t) can be written by the spin-flip mag± netic moments m± i = mix ± imiy and conjugate magnetic fields hi = hix ± ihiy as follows.  1

m+ (−q) h− (q, ω) + m− (−q) h+ (q, ω) H1 (t) = −L 2 q + mz (−q) hz (q, ω) e−i(ω+is)t . (4.69) ± Here m± (q) and h± (q, ω) are the Fourier transform of m± i and hi , respectively. The interaction (4.69) indicates that we can introduce a transverse response

4.3 Dynamical Susceptibility

127

m± (q, ω) to the time-dependent magnetic field h± (q, ω) exp(−i(ω + is)), instead of mα (q, ω) (α = x, y). The linear response relation is then written as follows. m± (q, ω) =



χ ± ∓ (q, ω + is)h± (q, ω).

(4.70)

γ

The transverse dynamical susceptibility χ ± ∓ (q, ω + is) is given by χ ± ∓ (q, ω + is) = L

i 



∞ 

0

 i(ω+is) ∓ m± dt. H (q, t), m (−q) e

(4.71)

The dynamical susceptibilities describe magnetic excitations. In order to understand the properties of χ − + (q, ω + is), for example, we express the time correlation function in (4.71) with use of the eigenstates |α and eigenvalues Eα of the Hamiltonian H . The dynamical susceptibility is then given as follows. χ

−+

  e−βEα  |γ |m− (q)|α|2 |γ |m+ (−q)|α|2 . (q, ω + is) = L − Z ω + is + Eγ − Eα ω + is − Eγ + Eα αγ (4.72)

Here β is the inverse temperature and Z denotes the partition function of the system. The spin-flip magnetic moment m− (q) is expressed as m− (q) =

2 † 2 − ak+q↓ ak↑ = Sqk . L L k

(4.73)

k

In the same way, m+ (q) is given by m+ (q) =

2 † 2 + ak+q↑ ak↓ = Sqk . L L k

(4.74)

k

− + has been introduced by (4.23), and Sqk is defined Note that the spin-flip operator Sqk

† + = ak+q↑ ak↓ . by Sqk The expression (4.72) indicates that the poles of χ − + (q, −ω − is) give the exci− }. The imaginary part of tation energies associated with the spin-flip processes {Sqk dynamical susceptibility describes the excitation spectra. In fact,

πL + M− (q, ω) − M− (q, −ω) . 

(4.75)

 e−βEα      γ m− (q)† α 2 δ(ω − ωγ α ), Z αγ

(4.76)

Im χ − + (q, ω + is) = Here M−+ (q, ω) =

128

4 Magnetic Excitations

M− (q, ω) =

 e−βEα      γ m− (q)α 2 δ(ω − ωγ α ), Z αγ

(4.77)

and ωγ α = (Eγ − Eα )/. We can verify the following relation by regarding exp(−βEα ) as exp(−βEγ + β(Eγ − Eα )) in (4.77). M− (q, ω) = eβω M−+ (q, −ω).

(4.78)

Substituting the above relation into (4.75), we find that Im χ − + (q, ω + is) =

πL 1 − e−βω M−+ (q, ω). 

(4.79)

This relation is known as the fluctuation-dissipation theorem. The same relation holds true for the other susceptibilities χ αγ (q, ω + is) (α, γ = x, y, z), where the correlation function M−+ (q, ω) at the r.h.s. of (4.79) should be replaced by M αγ (q, ω) =

 e−βEμ        μmα (q)ν ν mγ (−q)μ δ(ω − ωνμ ). Z μν

(4.80)

The following static spin fluctuation is also expressed by the dynamical susceptibility.      

 +    e−βEα   −  γ m (−q)α 2 +  γ m− (−q)† α 2 . m (q), m− (−q) + = Z αγ (4.81) In fact, the r.h.s. is expressed by M− (−q, ω) and M−+ (−q, ω) as follows.  ∞  

 + dω M− (−q, ω) + M−+ (−q, ω) . m (q), m− (−q) + = −∞

(4.82)

Making use of the relation (4.78) and the fluctuation-dissipation theorem (4.79), we obtain the relation between the static spin correlation function and the dynamical susceptibility as follows.  ∞   πL  + βω m (q), m− (−q) + = Im χ −+ (−q, ω + is). (4.83) dω coth  2 −∞ Finally, let us examine the static limit of the dynamical susceptibility. The relation (4.70) for q = 0 and ω = 0 is expressed as follows. mx (0, 0) − imy (0, 0) = χ −+ (0, 0)



1 hx (0, 0) − i hy (0, 0) . 2

(4.84)

In the static limit (ω → 0), χ − + (q, 0) is real according to the fluctuation dissipation theorem (4.79). Moreover in the limit q → 0, mα (q = 0, ω = 0) and

4.4 Dynamical Susceptibility in the RPA and Spin Wave Excitations

129

hα (q = 0, ω = 0) are real (see (4.61) and (4.62)). Thus (4.84) yields the following identity. 1 χ xx (0, 0) = χ yy (0, 0) = χ −+ (0, 0). 2

(4.85)

The dynamical susceptibilities are measured directly by means of the neutron magnetic scattering. The neutron has no charge but does have the spin sn = 1/2 and the magnetic moment μn = 1.913μn σ n , where μn is the nuclear magneton number defined by μn = e/2m0 c. m0 denotes the mass of neutron, and approximately equals that of proton mp (m0 = 1.00138mp ). σ n denotes the Pauli spin matrix for a neutron. When neutrons enter a solid, they interact with electrons via the magnetic dipole–dipole interaction as an electro-magnetic interaction. A neutron with wave vector k is then scattered to the state k  = k + q with the energy ε = 2 k 2 /2m0 and energy loss ω = 2 k 2 /2m0 − 2 k 2 /2m0 . Here q is called the scattering vector. The number of scattered neutrons per energy range ε around energy ε and per solid angle fraction Ω in the direction of k  is given as follows [14, 15].   2 k  d 2σ e2 2   f (q) = − 1.913 dΩ dε k me c2  L/π × (δαγ − qˆα qˆγ ) Im χ αγ (q, −ω + is). −βω 1 − e αγ

(4.86)

Here me is the mass of electron. f (q) is the magnetic form factor defined by f (q) =  (m) (m) dr ρi (r) exp(−iq · r), and ρi (r) is the normalized spin density defined by (m) the spin density on atom i as mi (r) = ρi (r) mi , mi being the total spin magnetic moment on atom i. Furthermore qˆ is the normalized scattering vector. Energy loss of a neutron by ω implies that magnetic excitations with energy ω exist in the solid. They are connected with spin fluctuations of solids via the fluctuation-dissipation theorem.

4.4 Dynamical Susceptibility in the RPA and Spin Wave Excitations We have shown in the last section that the dynamical susceptibility describes the magnetic excitations. We calculate here the dynamical susceptibility of the Hubbard model (4.26) using the equation of motion method combined with the Random Phase Approximation (RPA) [84]. We demonstrate that the spin-wave excitation spectra and the Stoner excitations are obtained from the poles of χ − + (q, −ω − is). We also discuss the numerical results of the spin wave stiffness constants in Fe and Ni in comparison with the experimental data.

130

4 Magnetic Excitations

In the equation of motion method, we express the dynamical susceptibility χ − + (q, ω + is), i.e., (4.71) by means of its retarded Green function χ − + (q, t) as χ

−+

 (q, ω + is) =

∞

−∞

χ − + (q, t) ei(ω+is)t dt.

(4.87)

The retarded Green function is defined by χ − + (q, t) = L

i + θ (t)[m− H (q, t), m (−q)]. 

(4.88)

 − † − Because of the relations m− (q) = (2/L) k Sqk and Sqk = ak+q↓ ak↑ , we can express χ − + (q, t) as a sum of the individual components χqk (t) as follows. 

χ − + (q, t) = 2

χqk (t).

(4.89)

k

Here χqk (t) =

i − θ (t)[SqkH (t), m+ (−q)]. 

(4.90)

The Fourier transform of (4.89) is given by χ − + (q, ω + is) = 2



χqk (ω + is),

(4.91)

χqk (t) ei(ω+is)t dt.

(4.92)

k

and  χqk (ω + is) =

∞

−∞

By differentiating (4.90) with respect to t, we obtain the equation of motion for χqk (t) as follows. i

∂χqk (t) − = −δ(t)[Sqk , m+ (−q)] ∂t i i i − + θ (t)[e  H t [Sqk , H ] e−  H t , m+ (−q)]. 

(4.93)

The commutation relation of the first term at the r.h.s. is given as follows.  −  2 Sqk , m+ (−q) = (nk+q↓ − nk↑ ). L

(4.94)

4.4 Dynamical Susceptibility in the RPA and Spin Wave Excitations

131

We have obtained in Sect. 4.2 the commutation relation in the second term at the r.h.s. of (4.93) (see (4.30) and (4.31)).

−  0  − − εk0 Sqk Sqk , H = − εk+q −

U  † † ak+q+q  ↓ ak† −q  σ  ak  σ  ak↑ − ak+q↓ ak† −q  σ  ak  σ  ak−q  ↑ . L    kqσ

(4.95) A difficulty arises from the interaction terms at the r.h.s. of the above equation − because they are not expressed by the operators {Sqk }. As we have mentioned in Sect. 4.2, the RPA simplifies these terms taking the diagonal part among various k  scattering terms and neglecting the other terms. For example, in the second term at the r.h.s. of (4.95), we make the RPA as follows.    †  † † ak+q+q  ↓ ak† −q  ↑ ak  ↑ ak↑ ≈ ak+q+q  ↓ ak↑ ak+q  ↑ ak↑ q

k

q

= −nk↑

 k

− Sqk 

 .

(4.96)

Making the same approximation to the other terms, we obtain the approximate form as  −

−  0 Sqk , H = − εk+q − εk0 Sqk −

−

 − U  U nk↑  − nk+q↓  nk  ↑  − nk  +q↓  Sqk + Sqk  . L  L  k

k

(4.97) Here we have replaced the number operators which appeared after the RPA with their average. Substituting (4.94) and (4.97) into (4.93), we obtain the equation of motion for − in a closed form. Sqk i



0 ∂χqk (t) 2 = −δ(t) nk+q↓  − nk↑  − εk+q − εk0 χqk (t) ∂t L

U  nk  ↑  − nk  +q↓  χqk (t) − L  k

+

 U nk↑  − nk+q↓  χqk  (t). L 

(4.98)

k

The above equation can be solved by the Fourier transform as follows. Multiplying exp(i(ω + is)t) in the above equation and integrating both sides with respect to t,

132

4 Magnetic Excitations

we find that 

U 0 ω + is + εk+q n↑  − n↓  χqk (ω + is) − εk0 + L

U

 2 nk↑  − nk+q↓  + nk↑  − nk+q↓  = χqk  (ω + is). L L 

(4.99)

k

Dividing both sides  by the coefficient of the l.h.s. and taking the sum with respect to k, we obtain k χqk (ω + is). Substituting the expression into (4.91), we obtain the dynamical susceptibility in the RPA. χ − + (q, ω + is) =

χ0− + (q, ω + is) 1−

U 4

χ0− + (q, ω + is)

.

(4.100)

Here χ0− + (q, ω + is) is the dynamical susceptibility for the noninteracting system given by χ0− + (q, ω + is) =

nk↑  − nk+q↓  4 . L ω + is + εk+q↓ − εk↑

(4.101)

k

As mentioned in the last section (see (4.72)), the poles of the dynamical susceptibility χ − + (q, −ω − is) give the excitation energies ωq associated with the † spin-flip excitations ak+q↓ ak↑ . According to the RPA susceptibility (4.100), the excitation spectra are obtained from the condition χ0− + (q, −ω − is) = ∞,

(4.102)

or 1−

U −+ χ (q, −ω − is) = 0. 4 0

(4.103)

The first equation (4.102) yields excitations due to the spin-flip processes of individual electrons, which are known as the Stoner excitations (see (4.36)). 0 − εk0 + U m. ωqk = εk+q↓ − εk↑ = εk+q

(4.104)

Another type of excitations is given by (4.103). As we analyzed in Sect. 4.2, (4.103) yields the spin wave excitation energies as well as the energies of the Stoner excitations in the strong ferromagnetic state at zero temperature (see (4.40)). ωq = Dq 2 .

(4.105)

Here D is the spin wave stiffness constant defined by (4.41). The wave functions of the excited states are not obtained in this method. The dynamical susceptibility is also useful for obtaining the free energy due to spin fluctuations as will be seen in the next chapter.

4.4 Dynamical Susceptibility in the RPA and Spin Wave Excitations

133

Detailed calculations of the spin wave stiffness constant D have been made on the level of the LDA + RPA. The results are rather sensitive to the details of the bands and the exchange splitting Δ. For bcc Fe, D = 280 meV Å2 was obtained for Δ = 1.94 eV [85] and is consistent with the value D = 280 meV Å2 obtained by the neutron scatterings [86] and 314 meV Å2 obtained by the low temperature behavior of the magnetization [87]. In the case of Ni, Wako et al. [88] obtained 285 meV Å2 for Δ = 0.47 eV, while the neutron experiments [89] show D = 395 ± 20 meV Å2 . The latter value is obtained theoretically when Δ = 0.40 eV is adopted [90].

Chapter 5

Spin Fluctuation Theory in Weak Ferromagnets

The single-site theory of magnetism allows us to understand the finite temperature magnetism qualitatively or semiquantitatively, starting from the microscopic Hamiltonian. We can analyze the magnetic properties from metals to insulators on the basis of the theory. The first-principles dynamical CPA can explain quantitatively high-temperature properties such as the Curie constant in the paramagnetic susceptibility. The single-site theory, however, neglects inter-site spin correlations. The latter influences the Curie temperature and the other quantities related with magnetic short range order. In particular, long-wave spin fluctuations are indispensable for understanding the weak ferromagnetism in ZrZn2 , Sc3 In, and Ni3 Al, which is characterized by a small Curie temperature (∼10 K) and a large Rhodes–Wohlfarth ratio (1). Here the Rhodes–Wohlfarth ratio is defined by the ratio of the observed Curie constant to the one based on the local moment model. In this chapter, we present a method which takes into account long-range intersite spin correlations at finite temperatures in the weak ferromagnets [91–96].

5.1 Free-Energy Formulation of the Stoner Theory We have described in Sect. 3.1 the Stoner theory in the weak Coulomb interaction region on the basis of the Hartree–Fock self-consistent equations, as well as the single-site spin fluctuation theories which go beyond the Stoner theory. In this section, we rederive the Stoner theory using the free energy at finite temperatures. We have constructed the dynamical CPA on the basis of the grand canonical ensemble in Chap. 3. There we treated the free energy F (μ, H, T ) for given chemical potential μ, external magnetic field H , and temperature T . Magnetization is then obtained from the thermodynamic relation as M =−

∂F (μ, H, T ) . ∂H

Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_5, © Springer-Verlag Berlin Heidelberg 2012

(5.1) 135

136

5

Spin Fluctuation Theory in Weak Ferromagnets

The uniform susceptibility is therefore obtained from the relation, χ =−

∂ 2 F (μ, H, T ) . ∂H 2

(5.2)

We can also make use of the free energy F (N, M, T ) for given electron number N , magnetization M, and temperature T when we discuss the thermodynamics of the system. In fact, the free energy F (μ, H, T ) is written as follows. e−βF (μ,H,T ) =



e−β(Eα (M, N )−μN −MH ) =



αN M

e−βF (N,H,T )+βμN .

(5.3)

N

Here we assumed that magnetization Mˆ commutes with the Hamiltonian Hˆ . Eα (M, N) is the energy eigen value when the electron number N and magnetization M are given. The free energy F (N, H, T ) is defined by e−βF (N, H, T ) =



e−β(Eα (M, N )−MH ) =

αM



e−βF (N, M, T )+βMH .

(5.4)

M

In the same way, the free energy F (N, M, T ) is given by e−βF (N,M,T ) =



e−βEα (M,N ) .

(5.5)

α

We can omit fluctuations of the macroscopic variables N and M, so that we obtain from (5.3) and (5.4) the relations F (μ, H, T ) = F (N, H, T ) − μN and F (N, H, T ) = F (N, M, T ) − MH as well as the associated thermodynamic relations due to stationary conditions. Therefore we find that F (N, M, T ) = F (μ, H, T ) + μN + MH.

(5.6)

This is a Legendre transformation from F (μ, H, T ) to F (N, M, T ). When we adopt the free energy F (N, M, T ) we can derive from (5.6) the following thermodynamic relation. H=

∂F (N, M, T ) . ∂M

(5.7)

The uniform susceptibility is therefore obtained by χ −1 =

∂ 2 F (N, M, T ) . ∂M 2

(5.8)

The Stoner theory, i.e., the Hartree–Fock theory can be reformulated with use of the free energy F (N, M, T ). In the following, we make use the notation m and h instead of M and H for convenience. The Hartree–Fock Hamiltonian to the Hubbard model has been obtained in (2.7). In the momentum representation, it is written as

5.1 Free-Energy Formulation of the Stoner Theory

follows. H˜ =

137

 

1 1  2 n − m2 . ε0 + U n − Δ · σ + εk nkσ − U 2 4 kσ

(5.9)

i

Here εk is the Fourier transform of the transfer matrix tij . Δ = U m/2 + h denotes an exchange splitting of the electron, n and m are the average electron number and the average magnetization per atom, respectively. The Hartree–Fock free energy F0 (μ, h, T ) per atom is then given by F0 (μ, h, T ) = −

1

1  ln 1 + e−β(ε˜0 +εk −μ−Δ·σ ) − U n2 − m2 . (5.10) βL 4 kσ

Here ε˜0 = ε0 + U n/2 − εF and μ = μ − εF , εF being the Fermi level in the nonmagnetic state. Introducing the density of states per atom per spin in the nonmagnetic state by ρ(ε) =

1 δ(ε − ε˜ 0 − εk ), L

(5.11)

k

the free energy F0 (μ, h, T ) is expressed as

1 F0 (μ, h, T ) = Φ0 (μ, h, T ) − U n2 − m2 , 4 

dε ρ(ε) ln 1 + e−β(ε−μ−Δ·σ ) . Φ0 (μ, h, T ) = −β −1

(5.12) (5.13)

σ

Here the electron number per site n and magnetization per site m have been written by n and m for convenience, so that ε˜0 and Δ stand for ε0 + U n/2 − εF and U m/2 + h, respectively. The stable configuration of electron number n and magnetization m under given μ and h is obtained by minimizing the free energy F0 (μ, h, T ); ∂F0 (μ, h, T )/∂n = 0 and ∂F0 (μ, h, T )/∂m = 0. These equations reproduce the Hartree–Fock equations (3.1) and (3.2); n = n and m = m. Here the average at the r.h.s. means the Hartree–Fock average. The Hartree–Fock free energy for given n, m, and T is obtained from the following Legendre transformation (see (5.6)). F0 (n, m, T ) = F0 (μ, h, T ) + μn + mh.

(5.14)

We can then verify the thermodynamic relations such as ∂F0 (n, m, T ) = μ, ∂n ∂F0 (n, m, T ) = h. ∂m

(5.15) (5.16)

138

5

Spin Fluctuation Theory in Weak Ferromagnets

The explicit form of the free energy F0 (n, m, T ) is calculated as follows. One electron part of the free energy (5.13), i.e., Φ0 (μ, h, T ) is expressed by μ and Δ. μ is expanded by Δ according to the Hartree–Fock equation n = n (see (3.5)).    c 2 2 2 (5.17) μ = a0 Δ 1 + a + 3b + Δ + ··· . a Here a0 , a, and b are defined by (3.7) and (3.6). Therefore we can expand Φ0 (μ, h, T ) given by (5.13) with respect to Δ as follows.     1 Φ0 (μ, h, T ) = Φ0 (μ, h, T ) m=0 − nμ − a0 Δ2 + a0 a 2 + b Δ4 2 + ··· .

(5.18)

Thus we obtain Φ0 (n, m, T ) ≡ Φ0 (μ, h, T ) + μn + mh   1 = Φ0 (n, m, T ) m=0 + mΔ − U m2 − a0 Δ2 2   1 2 + a0 a + b Δ4 + · · · . 2

(5.19)

On the other hand, with use of the Hartree–Fock equation m = m, we can expand m with respect to Δ as follows, as has been shown in (3.9). 

 (5.20) m = 2a0 Δ 1 − 2a 2 + b Δ2 + · · · . Solving (5.20) with respect to Δ, we find Δ=

1 2a 2 + b 3 m+ m + ··· . 2a0 8a03

(5.21)

From (5.12), (5.14), (5.19), and (5.21) we obtain the free energy F0 (n, m, T ) expanded by m as follows. F0 (n, m, T ) = F0 (n, 0, T ) +

1 1 m2 + g(T ) m4 + · · · . 2χHF 64

(5.22)

Here F0 (n, 0, T ) is the Hartree–Fock free energy in the nonmagnetic state given by 

1 2 F0 (n, 0, T ) = nεF − (5.23) dε ρ(ε) ln 1 + e−βε − U n2 . β 4 The coefficient χHF in the 2nd order term in (5.22) is given by   1 1 1 U 1 = −U = − . χHF 2 a0 χ0 2

(5.24)

5.1 Free-Energy Formulation of the Stoner Theory

139

The coefficient g(T ) in the 4th-order term of (5.22) is given by g(T ) =

2(2a 2 + b) a03

.

(5.25)

Note that the coefficient a0 in (5.24) and (5.25) depends on temperature T via the Fermi distribution function and the Fermi level εF . It has been obtained before, as shown in (3.14),  π2 RT 2 + · · · , (5.26) a0 (T ) = ρ 1 − 6 and R is defined by (3.15); R = (ρ (1) /ρ)2 − ρ (2) /ρ. Here ρ, ρ (1) , and ρ (2) denote ρ(0), ρ (1) (0), and ρ (2) (0), respectively. The coefficient g(T ) in the 4th-order term is obtained in the same way as   π2 F1 1+ RT 2 + · · · , (5.27) g(T ) = ρ 6 and F1 = (ρ (1) /ρ)2 − ρ (2) /3ρ. Equation (5.22) is the Hartree–Fock free energy for given n, m, and T in weak ferromagnets. From the thermodynamical relation (5.8), we find that χHF given by (5.24) is the uniform susceptibility in the Hartree–Fock approximation. It agrees with (3.11). The parameter χ0 denotes the susceptibility for a noninteracting system. We obtain from the susceptibility (5.24) the Curie temperature in the Stoner theory as follows.  6(ρU − 1) , (5.28) TC = π 2 RρU which agrees with the expression (3.16). The spontaneous magnetization m is obtained from the thermodynamical relation (5.7) with h = 0. 1 1 ∂F0 (n, m, T ) = m + g(T )m3 + · · · = 0. ∂m χHF (T ) 16

(5.29)

For small magnetization we obtain  m(T ) =

−

16 . χHF (T )g(T )

This is equivalent to (3.21). Therefore we can derive (3.22) again:   2 T m(T ) = m(0) 1 − . TC

(5.30)

(5.31)

140

5

Spin Fluctuation Theory in Weak Ferromagnets

√ Here  the ground-state magnetization m(0) is given by (3.10); m(0) = 8(ρU − 1)/ ρU F1 U 2 . We have rederived from the Hartree–Fock free energy (5.22) all the results of the Stoner theory obtained in Sect. 3.1. The free energy (5.22) provides us with a starting point to take into account the nonlocal spin fluctuations on the basis of the free energy.

5.2 Self-Consistent Renormalization Theory The Hartree–Fock free energy which we derived in the last section is valid only for a small U limit. It overestimates the Curie temperature and does not lead to the Curie–Weiss law in the paramagnetic susceptibility as we have mentioned in Sects. 3.1 and 5.1. In this section, we express the free energy with use of the dynamical susceptibility, and take into account the higher-order contribution in U which yields the nonlocal spin fluctuations [91, 92]. We adopt the Hubbard model in the momentum representation as follows (see (4.26)–(4.28)). H = H0 + HI ,  εk0 nkσ , H0 =

(5.32) (5.33)

kσ

HI =

U  † ak1 +qσ ak†2 −q−σ ak2 −σ ak1 σ . 2L

(5.34)

k1 k2 qσ

Here εk0 = ε0 + εk is the one-electron energy for noninteracting Hamiltonian. L denotes the number of the lattice points. The free energy F (μ, h, T ) for the grand canonical ensemble is given by

F (μ, h, T ) = −β −1 Tr e−β(H −μN ) .

(5.35)

Taking the derivative of F with respect to the interaction parameter U , we have ∂F (μ, h, T )/∂U = HI /U . Integration of both sides with respect to parameter U leads to the following relation. F (μ, h, T ) = −β

−1

Tr e−β(H0 −μN ) +



U

dU 0





HI U

 U

.

(5.36)

Here ∼U  denotes the thermal average for the Hamiltonian with coupling parameter U  . In the Hartree–Fock approximation, we replace the thermal average ∼U  with U that of a Hamiltonian for independent particle system ∼0 , so that 0 dU  ·

5.2 Self-Consistent Renormalization Theory

141

HI /U  U  ≈ LU n↑ 0 n↓ 0 . Therefore we can express the free energy (5.36) as 1 L

F (μ, h, T ) = F0 (μ, h, T ) +



U

dU 



0

HI U

 U

 −

HI U

  .

(5.37)

0

Here we redefined F (μ, h, T ) as the free energy per site. F0 (μ, h, T ) is the free energy per site in the Hartree–Fock approximation and was given by (5.10) in the last section. F0 (μ, h, T ) = −

1

1  ln 1 + e−β(ε˜0 +εk −μ−Δ·σ ) − U n2 − m2 . βL 4

(5.38)

kσ

The free energy for given electron number per site n, magnetization per site m, and the temperature T is then obtained by the Legendre transformation (5.6): F (n, m, T ) = F (μ, h, T ) + μn + mh.

(5.39)

Substituting (5.37) into (5.39), we obtain F (n, m, T ) = F0 (n, m, T ) + F.

(5.40)

Here F0 (n, m, T ) is the free energy in the Hartree–Fock approximation defined by (5.14), F0 (n, m, T ) = F0 (μ, h, T ) + μn + mh,

(5.41)

and F is the correlation correction to the Hartree–Fock free energy. 1 F = L



U

dU





0

HI U





HI − U U

  .

(5.42)

0

To calculate the interaction part F in (5.40), we express the interaction HI by  † ak↓ and m− (q) = means of the spin-fluctuation operators m+ (q) = (2/L) k ak+q↑  † (2/L) k ak+q↓ ak↑ as follows.   1 1 m+ (q), m− (−q) + . HI = U N − LU 2 8 q

(5.43)

Substituting (5.43) into (5.42), we obtain F = −

1 8

 0

U

dU 

       

m+ (q), m− (−q) + − m+ (q), m− (−q) + 0 . q

(5.44) The spin fluctuation terms are given by dynamical susceptibility according to the relation (4.83).

142

5

Spin Fluctuation Theory in Weak Ferromagnets

Fig. 5.1 The contour C0 (left), contour C (center), and contour Cε on the complex plane. Note that the contour C0 is chosen to pass just on the origin of the complex plane

 +    m (q), m− (−q) + = πL



∞

−∞

dω coth

βω Im χ −+ (−q, ω + is). 2

(5.45)

Thus, we can express the interaction part by means of the dynamical susceptibility as follows.  ∞  U βω   dU dω coth F = − 8πL 0 2 −∞  −+

Im χ (q, −ω − is) − χ0−+ (q, −ω − is) . (5.46) × q

Here we made the transformation q → −q, ω → −ω and used the relation Imχ −+ (q, −ω + is) = −Imχ −+ (q, −ω − is) for convenience. The integral of the dynamical susceptibility with respect to ω in the interaction (5.46) is expressed by a sum of Matsubara frequencies on the imaginary axis as follows.  ∞ ∞ βω 1  −+  −+ Im χ (q, ω + is) = dω coth χ (q, iωn ). (5.47) I= 2π −∞ 2 β n=−∞ Here ωn = 2πn/β. In order to verify the above relation we rewrite the integral at the l.h.s. as  ∞  βω −+ χ (q, ω + is) I= dω coth 4πi −∞ 2  −∞ βω −+ dω coth (5.48) + χ (q, ω − is) . 2 ∞ These are expressed as an integral along the contour C0 around the real axis as shown in Fig. 5.1. Note that the integral should pass the origin because the function coth βω/2 is singular there. We can consider the contour C0 as the superposition of the contours C and Cε . The contour C can be changed to C , the sum of the contour around the upper imaginary axis and that around the lower imaginary axis as shown

5.2 Self-Consistent Renormalization Theory

143

Fig. 5.2 The contour C on the complex plane and singular points zn = iωn on the imaginary axis leading to the residues

in Fig. 5.2. Because the susceptibility is analytic on the upper and lower complex planes, the contributions from C and Cε are calculated by summing up all residues of coth βω/2 at zn = iωn along the imaginary axis, so that we reach the relation (5.47). Making use of the formula (5.47), we obtain alternative expression of F .  U 

1 dU  (5.49) χ −+ (q, −iωn ) − χ0−+ (q, −iωn ) . F = − 4βL 0 qn For the calculation of F , we need an explicit form of the dynamical susceptibility. The general form of the dynamical susceptibility is written as follows. χ − + (q, ω + is) =

χ0− + (q, ω + is)

1−

U 4

χ0− + (q, ω + is) + λ(q, ω + is)

.

(5.50)

χ0− + (q, ω + is) is the dynamical susceptibility in the Hartree–Fock approximation given by (4.101). The second term in the denominator is the RPA correction (see (4.100)), and the last term λ(q, ω + is) describes corrections beyond the RPA. The RPA susceptibility is insufficient to describe finite temperature properties, though it explained the spin wave excitations at zero temperature as discussed in Sect. 4.4. We assume here that λ(q, ω + is) = 0 at zero temperature, but that λ(q, ω + is) ≈ λ(0, 0) = λ(T ) at finite temperatures, expecting low energy excitations and fluctuations around q = 0 to be important for the weak ferromagnetism. We also assume λ(T )  1. Finally, we adopt the following dynamical susceptibility. χ − + (q, ω + is) =

χ0− + (q, ω + is)

1−

U 4

χ0− + (q, ω + is) + λ(T )

.

(5.51)

The Hartree–Fock dynamical susceptibility is given as follows as we have obtained in (4.101). χ0− + (q, ω + is) =

4  f (εk − Δ) − f (εk+q + Δ) . L ω + is + εk+q − εk + 2Δ k

(5.52)

144

5

Spin Fluctuation Theory in Weak Ferromagnets

Here f (ω) is the Fermi distribution function, Δ = U m/2 + h, and we redefined εk as εk = εk0 − μ + U n/2. εk0 is the one-electron energy eigen value for noninteracting electrons. Substituting (5.51) into (5.49) and performing the integral on parameter U  , we obtain    U −+ 1  U −+ ln 1 − χ0 (q, −iωn ) + λ − χ0 (q, −iωn ) . (5.53) F = βL qn 4 4 Here we have neglected an additional term ln(1 + λ), assuming that |λ|  1. The final form of the free energy for given n, m, and T is expressed by (5.40), (5.41), and (5.53). We have obtained in the last section an explicit form of F0 (n, m, T ) (see (5.22)). The free energy correction (5.53) is given as a function of Δ, therefore we can regard F as a function of m. The correction λ(T ) due to thermal spin fluctuations has not yet been determined. In order to determine λ(T ) we make use of a sum rule in the static limit. In the paramagnetic state, the system is magnetically uniform. Therefore χ xx = yy χ = χ zz = χ . According to the relation (4.85), this implies that 1 χ = χ −+ (q = 0, ω = 0). 2

(5.54)

For a noninteracting limit (i.e., the Hartree–Fock limit), we have 1 χ0 = χ0−+ (q = 0, ω = 0). 2

(5.55)

Below TC , the relation χ xx = χ yy = χ zz is no longer satisfied because the z direction is not magnetically equivalent to the other directions. Instead, χ xx = χ yy = χ −+ (q = 0, ω = 0)/2 should diverge below TC because the Hamiltonian is rotationally invariant. For the susceptibility (5.51), this implies that 1−

1 −+ (q = 0, ω = 0) + λm = 0. U χm0 4

(5.56)

Here the subscript m means that the magnetization m is finite, and it is assumed to be given in the above equation. In the following, we consider the paramagnetic state for simplicity. Substituting (5.51) and (5.55) into (5.54), we obtain χ=

χ0

,

(5.57)

1 1 λ 1 = − U+ . χ χ0 2 χ0

(5.58)

1−

1 2U

χ0 + λ

i.e.,

5.2 Self-Consistent Renormalization Theory

145

On the other hand, the uniform susceptibility is obtained from the thermodynamic relation of the free energy via (5.8) independently of the sum rule (5.58). 1 = χ



∂ 2 F (n, m, T ) ∂m2



 ∂ 2 F0 (n, m, T ) ∂m2 m=0   2 ∂ F (n, m, T ) + . ∂m2 m=0 

= m=0

(5.59)

Here we added the subscript m = 0 because of being in the paramagnetic state. The Hartree–Fock part at the r.h.s. is obtained from the expression of the free energy (5.22) as 

∂ 2 F0 (n, m, T ) ∂m2

 = m=0

1 1 1 = − U. χHF χ0 2

(5.60)

Therefore the susceptibility (5.59) is expressed as   2 1 1 ∂ F (n, m, T ) 1 = − U+ . χ χ0 2 ∂m2 m=0

(5.61)

From the sum rule (5.58) and the thermodynamic susceptibility (5.61), we obtain the self-consistent equation to determine the spin fluctuation parameter λ(T ) as  λ(T ) = χ0

∂ 2 F (n, m, T ) ∂m2

 .

(5.62)

m=0

The r.h.s. is obtained from (5.53) as  χ0− + (q, −iωn ) − 4λ ∂ 2 χ0− + (q, −iωn ) ∂ 2 F (n, m, T ) U2  U = − 16βL qn 1 − U4 χ0− + (q, −iωn ) + λ ∂m2 ∂m2 +

1 (1 −

U 4

χ0− + (q, −iωn ) + λ)2



∂χ0− + (q, −iωn ) ∂m

2 . (5.63)

Solving the self-consistent equation (5.62), we obtain λ(T ) at each temperature T , thus obtain the susceptibility χ via (5.57) or (5.58). This self-consistent scheme based on the RPA is known as the self-consistent renormalization theory (SCR) [93]. The Curie temperature TC is obtained from (5.57) with the condition 1/χ = 0. 1−

1 U χ0 (TC ) + λ(TC ) = 0. 2

(5.64)

The above equation shows that the RPA (λ = 0) does not improve the Curie temperature of the Hartree–Fock approximation. In the SCR, λ > 0 at finite temperatures,

146

5

Spin Fluctuation Theory in Weak Ferromagnets

Fig. 5.3 The Curie temperature in the SCR theory (solid line) and the Hartree–Fock theory (dashed line) as a function of the Stoner parameter α = U χ0 (T = 0)/2 [91, 92]

therefore the renormalization λ can reduce TC . According to the numerical calculations based on the free electron gas model, the Curie temperature obtained by (5.64) reduces the Hartree–Fock (Stoner) TC by a factor of 10 (see Fig. 5.3). Since the single-site spin and charge fluctuations reduce the Hartree–Fock TC by a factor of 5 according to Table 3.2 in the last section, the results suggest that the nonlocal spin fluctuations can reduce TC by a factor of 2. But the theory based on the realistic Hamiltonian which allows us to make quantitative calculations of TC has not yet been developed. Equation (5.64) is equivalent to (5.58) with χ −1 = 0. 0=

1 λ(TC ) 1 − U+ . χ0 (TC ) 2 χ0 (TC )

(5.65)

Taking the difference between (5.58) and (5.65) near TC , we find λ(T ) − λ(TC ) 1 = . χ χ0 (TC )

(5.66)

Expanding λ(T ) with respect to T near TC , we find the Curie–Weiss law. λ(TC ) 1 = (T − TC ). χ χ0 (TC )

(5.67)

The expression above indicates that the Curie–Weiss law is caused by the selfconsistent renormalization of spin fluctuations beyond the Hartree–Fock approximation. The numerical calculations verify that the susceptibilities approximately follow the Curie–Weiss law as shown in Fig. 5.4. It should be noted that the Curie

5.2 Self-Consistent Renormalization Theory

147

Fig. 5.4 Inverse susceptibility vs. temperature curves for various Stoner parameters α = U χ0 (T = 0)/2. Note that TC = 0.005, 0.010, 0.015 for α = 1.008, 1.021, and 1.036, respectively. The curve in the Stoner theory is also presented for TC = 0.01 [91, 92]

constant is determined by the spin fluctuations in the long-wave regime as seen from (5.67). The theory explains the small TC and the large Rhodes–Wohlfarth ratio in the weak ferromagnets such as ZrZn2 and Sc3 In. The same type of theory was also developed by Cyrot [94], and Hertz and Klenin [95, 96]. Cyrot took into account Onsager’s reaction field in the calculation of the susceptibility. The spin correlation function was calculated with use of the fluctuation-dissipation theorem. In the theory by Hertz and Klenin, they started from the variational principle for the free energy in the functional integral method, and adopted the RPA type trial functional. Making use of the self-consistent phonon approximation, they obtained the free energy. Both theories yield essentially the same results as those obtained by the self-consistent renormalization theory of the weak ferromagnetism in metals.

Chapter 6

Antiferromagnetism and Spin Density Waves

As we have mentioned in Sect. 1.7, magnetic metals and compounds show a variety of magnetic structures. These include the ferromagnetic structure, the antiferromagnetic (AF) structure, the spin density waves (SDW), and more complex magnetic structures. In this chapter we clarify the microscopic mechanism for the formation of the antiferromagnetism as well as the SDW, and describe theoretical approaches to the complex magnetic structures in metals. We treat in Sect. 6.1 the antiferromagnetism at half filling with use of the Hartree–Fock approximation and the dynamical CPA, and argue the stability of the AF on the U –n plane. In Sect. 6.2, we derive the generalized static susceptibility, and explain the stability of the antiferromagnetism as well as the SDW in 3d transition metals. The susceptibility is not useful for understanding more complex magnetic structures in the itinerant electron system. We present in Sect. 6.3 the molecular dynamics approach which automatically finds the stable magnetic structure at finite temperatures. In the last section, we present a phenomenological theory which is useful for understanding complex magnetic structures in weak itinerant electron magnets, and discuss the multiple spin density waves (MSDW).

6.1 Antiferromagnetism in Metals The 3d transition metals in the vicinity of the middle on the periodic table and their alloys show various antiferromagnetic states. The bcc-base Cr alloys with a few percent of Mn, for example, show the antiferromagnetic state with a simple cubic sublattice as shown in Fig. 6.1. The fcc-base Mn alloys also show the antiferromagnetic structure of the first-kind in which the (001) ferromagnetic plane alternatively changes the magnetization in direction along the c-axis (see Fig. 1.10(c)). The αMn shows more complex noncollinear antiferromagnetic structure in which 29 Mn atoms in the primitive unit cell have magnetic moments different in size and direction [97]. We present in this section the Hartree–Fock theory of the antiferromagnetism at half filling on the basis of the Hubbard model, and clarify the so-called nesting mechanism for the formation of the antiferromagnetism. Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_6, © Springer-Verlag Berlin Heidelberg 2012

149

150

6 Antiferromagnetism and Spin Density Waves

Fig. 6.1 Magnetic phase diagram of Cr–V and Cr–Mn alloys showing the longitudinal spin density wave state (LSDW), the transverse spin density wave state (TSDW), the antiferromagnetic state (AF), and the paramagnetic state (P) [97]

The Hubbard Hamiltonian in the Hartree–Fock approximation is written as (see (2.7)) H=



† (H σ )ij aiσ aj σ −



ij σ

U ni↑ ni↓ .

(6.1)

i

Here (H σ )ij = εiσ δij + tij (1 − δij ) and εiσ = ε0 + U ni /2 − U mi σ/2. The selfconsistent local charge and magnetic moment on site i are then given by  ni  =

dω f (ω − μ)

ρiσ (ω),

(6.2)

σρiσ (ω).

(6.3)

σ

 mi  =



dω f (ω − μ)

 σ

Here f (ω) is the Fermi distribution function, μ is the chemical potential, and ρiσ (ω) is the density of states on site i and spin σ in the Hartree–Fock approximation. The latter is given by the Green function Giiσ (z) as follows. 1 Im Giiσ (z), π  i|κσ κ|iσ   Giiσ (z) = = (z − H σ )−1 ii . z − εκσ κ ρiσ (ω) = −

(6.4) (6.5)

6.1 Antiferromagnetism in Metals

151

Here z = ω + iδ with δ being the infinitesimal positive number, εκσ is the Hartree– Fock one-electron energy eigen value, and the eigen state is assumed to be given by  φκσ = i φiσ i|κσ . We consider here the antiferromagnetic (AF) state with two sublattices η = ± on the simple cubic (sc) lattice with the lattice constant a (see Fig. 1.10(a)) for simplicity. The following arguments are also applicable to the AF structure on the bcc lattice (see Fig. 1.10(b)). Note that each sublattice forms the fcc lattice in the case of the sc structure. We adopt the unit cell with 2 atoms, one with an up magnetic moment belonging to the sublattice (+), and another with a down magnetic moment belonging to the (−) sublattice. The primitive translational vectors are given by a = a(1, 1, 0), b = a(0, 1, 1), and c = a(1, 0, 1). The volume of the unit cell is 2a 3 . These vectors form the fcc lattice with lattice constant 2a. The Bloch functions |kη are given by 1  φ(r − R l − η± ) e−ik·(Rl +η± ) . L/2 l fcc

|k± = √

(6.6)

Here k denotes a wave vector in the first Brillouin zone of the fcc lattice. L denotes the number of simple-cubic lattice points. The atomic position vectors η± are given by η+ = 0 and η− = a(1, 1, 1), respectively. The atomic levels belonging to the sublattice η(= ±1) are given by εησ = ε0 + U n/2 − U |m|ησ/2. When there is no electron hopping between the atoms in the same sublattice and only the nearest-neighbor electron hoppings exist between different sublattices, we have the following 2 × 2 Hamiltonian matrix.       εk ε+σ   kη Hσ kη = . (6.7) εk ε−σ  Here εk reduces to the energy eigenvalue for simple cubic lattice; εk = t sc i exp(ik · Ri ) = −2|t|(cos kx a + cos ky a + cos kz a), t being the nearest neighbor hopping matrix element. The eigenfunction is expanded by |kη as  ψkνσ = |kη uηνσ (k). (6.8) η

Solving the 2 × 2 eigenvalue problem for uηνσ (k) we have the eigen values ενσ (k) as follows. ' ενσ (k) = ε 0 + ν Δ2 + εk2 . (6.9) Here ν takes ±. ε0 = (ε+σ + ε−σ )/2 = ε0 + U n/2 and Δ = U |m|/2. Note that k is a point in the Brillouin zone (BZ) of the fcc sublattice. The eigen vectors uηνσ (k) are given by   Δ 1 2 . (6.10) 1 − ηνσ ' uηνσ (k) = 2 Δ2 + εk2

152

6 Antiferromagnetism and Spin Density Waves

Note that the Brillouin zone (BZ) of the sc lattice with lattice constant a is twice as large as that of the fcc. We can express the eigen values using all the k points in the BZ of the sc lattice. In fact, any k  point of the BZ for the sc lattice which is outside the BZ of the fcc lattice is expressed as k  = k + Q by using a k point in the fcc sublattice and a Q vector such that Q = (π/a)(±1, ±1, ±1), resulting with the relation εk  = εk+Q = −εk . Therefore ' when the eigen value inside the BZ of the

fcc lattice is given by ε0 + (εk /|εk |) Δ2 + εk2 , another eigen value on the same k ' point is given as ε 0 + (εk  /|εk  |) Δ2 + εk2 with use of the k  = k + Q point in the BZ of the sc lattice. This means that the energy eigenvalue is expressed by using the k points of the BZ of the sc lattice as ' εk εσ (k) = ε 0 + Δ2 + εk2 . (6.11) |εk | Its eigenfunction is expressed by (6.8) in which uηνσ (k) has been replaced by uησ (k) such that   εk Δ 1 ' . (6.12) 1 − ησ uησ (k)2 = 2 |εk | Δ2 + ε 2 k

The Green function on the (+) sublattice is given by 2  u+σ (k)2 = L z − εσ (k) sc

(+) Giiσ (z) =

k



1 − sgn(ε)σ √ Δ2 2 Δ +ε . (6.13) dε ρ(ε) √ z − ε 0 − sgn(ε) Δ2 + ε 2

Here ρ(ε) is the density of states per atom per spin for the noninteracting sc system. It is worth mentioning that the Green function is alternatively written as

 z − ε−σ ρ(ε) dε (+)  Giiσ (z) = , (6.14) z − εσ (z − εσ )(z − ε−σ ) − ε where ε−σ = ε 0 + εσ is the Hartree–Fock potential on the (+) sublattice. (+) (+) The density of states ρσ (ε) on a (+) sublattice site is given by ρσ (ε) = (+) −π −1 ImGiiσ (z) according to the formula (6.4). With use of (6.2), (6.3), and (6.13), we obtain the local charge and local magnetic moment on the (+) sublattice as follows.  ( )  n = 2 dε ρ(ε)f ε 0 − μ + sgn(ε) Δ2 + ε 2 , (6.15)  m = 2

) (  sgn(ε) dε ρ(ε) √ f ε 0 − μ + sgn(ε) Δ2 + ε 2 . Δ2 + ε 2

(6.16)

It should be noted that the self-consistent equations (6.15) and (6.16) can be derived for any antiferromagnets with two sublattices in which there is no electron hopping between atoms belonging to the same sublattice. For example, in the

6.1 Antiferromagnetism in Metals

153

case of the AF on the bcc lattice as shown in Fig. 1.10(b), each sublattice forms a simple cubic lattice, so that the lattice sum in the Bloch function (6.6) is taken over the sc lattice instead of the fcc one. The atomic position vectors in this case are defined by η+ = 0 and η− = (a/2)(1, 1, 1). The energy eigen value εk in (6.7)  should be read as εk = t bcc i exp(ik·Ri ) = −8|t| cos kx a/2 cos ky a/2 cos kz a/2. The BZ of the sc sublattice is expanded to that of the bcc lattice by Q vectors (2π/a)(±1, 0, 0), (2π/a)(0, ±1, 0), and (2π/a)(0, 0, ±1). Thus, taking the same steps, we reach (6.15) and (6.16). At half-filling and T = 0, the chemical potential satisfying (6.15) is given by μ = ε 0 . The self-consistent equation (6.16) reduces to the following one. 1 = U



W/2 0

ρ(ε) dε . √ Δ2 + ε 2

(6.17)

Here W denotes the band√width and we made use of the relation ρ(ε) = ρ(−ε). Note that the function 1/ Δ2 + ε 2 is positive and monotonically increases with decreasing Δ = U m/2. Consequently, the  ∞r.h.s. of (6.17) also monotonically increases, and diverges at Δ = 0 because 0 dε ρ(ε)/|ε| ∼ −ρ(0) ln |0|. Thus the self-consistent equation (6.17) has a nonzero solution irrespective of the Coulomb interaction energy strength U ; the antiferromagnetic state exists at half-filling for any finite value of U (> 0). Equation (6.17) has the same form as the BCS gap equation for the super conducting state. There the band width W/2 is replaced by the Debye cut-off frequency, the Coulomb interaction parameter U is replaced by the electron–electron attractive interaction V caused by the electron–phonon interaction, and Δ corresponds to the super-conducting order parameter. When U  W/2, we have Δ  W/2. Thus the integral of the r.h.s. of (6.17) is evaluated as  W/2    W/2 W/2 ρ(ε) dε ρ(ε) dε ρ(ε) dε ' + ≈ ρ(0) + ρ(0) ln . ≈ Δ |ε| Δ 0 0  Δ2 + εk2 (6.18) We then obtain the ground-state sublattice magnetization as m =

1 W − ρ(0)U e . U

(6.19)

In the case of the completely flat band ρ = 1/W , one can directly perform the integral in (6.17) and obtain the sublattice magnetization as m =

W/U . sinh W/U

(6.20)

For large W/U , we obtain m = (2W/U ) exp(−1/ρ(0)U ), which is essentially the same as (6.19) though the prefactor differs from that of (6.19) by a factor of two.

154

6 Antiferromagnetism and Spin Density Waves

It should be noted that the antiferromagnetism at half-filling mentioned above is accompanied by a gap 2Δ on the Fermi surface according to (6.11). The kinetic energy gain due to the formation of the gap is the origin of the antiferromagnetism. This type of the AF is known as the gap-type antiferromagnetism or the nestingtype antiferromagnetism because the gap is caused by a nesting of the energy eigen values which are connected each other via Q vectors. At finite temperatures, the chemical potential μ = ε 0 satisfies (6.15). Using the relation f (−ε) − f (ε) = tanh(βε/2), the self-consistent equation (6.16) is written as √  W/2 1 ρ(ε) dε β Δ2 + ε 2 = . (6.21) tanh √ U 2 Δ2 + ε 2 0 The Néel temperature TN at which the antiferromagnetism disappears is obtained from the condition Δ = 0 in the above equation. 1 = U



W/2

dε 0

ε ρ(ε) tanh . ε 2TN

(6.22)

Here the r.h.s. is approximately obtained by replacing the DOS with its value at the Fermi level and making integration by parts as follows. 

W/4TN

ρ(0) 0

   W/4TN tanh x W/4TN 2 ≈ ρ(0) [ln x tanh x]0 dx − ln x sech x . x 0 (6.23)

When W/4TN  1 (i.e., the case of weak antiferromagnet), we can replace the upper bound of the integral in the second term with the infinity and make use of the for∞ mula − 0 ln x sech2 x dx = ln 4γ /π . Here ln γ is Euler’s constant ln γ ≈ 0.5772. Therefore (6.22) is written as 1/U = ρ(0) ln γ W/πTN . Solving the equation with respect to TN we obtain TN =

1 γ W − ρ(0)U W − 1 ≈ 1.13 e ρ(0)U . e π 2

(6.24)

The ground-state magnetization (6.19) and the Néel temperature (6.24) are applicable in the small U limit because the Hartree–Fock approximation is exact in the limit. The situation is quite different from the ferromagnetic case. There the Hartree–Fock approximation predicted that the magnetic state appears in the region ρ(0)U > 1 where the approximation is not justified in general. When the Coulomb interaction becomes large, the expression (6.24) of TN is no longer applicable. For larger values of U , we can apply the single-site theory at finite temperatures presented in Chap. 3. Let us adopt the static approximation to the functional integral method for simplicity. Introducing the site-dependent coherent (±) potential Σiσ (z) = Σσ (z), where (+) or (−) denotes the type of sublattice on site i, into the potential part of the free energy (3.67), we expand the correction terms

6.1 Antiferromagnetism in Metals

155

with respect to the site. We reach again the free energy (3.82) in the SSA as well as the CPA equation (3.85). In the antiferromagnetic (AF) state with two sublattices, we have a symmetric (−) (+) relation Σσ (z) = Σ−σ (z) for the coherent potential. Accordingly we have the re(−) (+) (±) lations Fσ (z) = F−σ (z) for the coherent Green function Fiσ (z) = Fσ (z) and (−) (+) (±) E (ξ ) = E (−ξ ) for the impurity energy Ei (ξ ) = E (ξ ). Thus, the free energy (3.67) as well as the other physical quantities can be expressed by those on the (+) sublattice. The free energy per site is finally written as  βU (+) −1 ˜ (6.25) FCPA = F − β ln dξ e−βE (ξ ) , 4π and the CPA equation (3.85) for the AF state is expressed as Gσ(+) (z, ξ ) = Fσ(+) (z).

(6.26)

(+)

Here Gσ (z, ξ ) is the impurity Green function (3.86) on the (+) sublattice. As seen from (6.14), the coherent Green function is expressed by * + + z − Σ (+) (z)  ρ(ε) dε −σ (+) ' . (6.27) Fσ (z) = , (+) (+) (+) z − Σσ (z) (z − Σ+ (z))(z − Σ− (z)) − ε Figure 6.2 shows the calculated Néel temperatures as a function of the Coulomb interaction parameter U for the half-filled band Hubbard model [53]. The results of the quantum Monte-Carlo method are also shown there. The density of states √ for the noninteracting state is assumed to be semielliptical: ρ(ε) = (2/πW 2 ) W 2 − ε 2 , which is known to be realized on the Bethe lattice in infinite dimensions. In the weak interaction regime, the Néel temperature (TN ) exponentially increases with increasing U in accordance with the Hartree–Fock formula (6.24). It shows the maximum around U/W = 2. When U becomes stronger, the Néel temperature approaches to the result of the molecular-field approximation in the Heisenberg model with  the super exchange interaction, i.e., H = − (i,j ) Jij S i · Sj with Jij = −4|tij |2 /U (see (1.86)). In the case of the Bethe lattice in infinite dimensions it is given by TN = W 2 /4U assuming the nearest-neighbor electron hopping. In finite dimensions, this should be reduced by the magnetic short-range order. It is obtained accurately by the high-temperature expansion. Note that there is the metal–insulator crossover above TN in general, so that the paramagnetic metal (PM) changes to the paramagnetic insulator (PI) with increasing Coulomb interaction (see Fig. 6.2). When the electron number is deviated from the half-filling, one has to solve both equations (6.15) and (6.16) self-consistently. In general the kinetic energy gain due to the gap formation becomes smaller as the electron number deviates from n = 1, and the antiferromagnetism becomes unstable. Figure 6.3 shows a typical magnetic phase diagram obtained by the Gutzwiller approximation to the Hubbard model [98]. For small Coulomb interactions, there is the transition from the AF to the para-

156

6 Antiferromagnetism and Spin Density Waves

Fig. 6.2 Néel temperature (TN ) vs. Coulomb interaction curves in the half-filled Hubbard model on the Bethe lattice in infinite dimensions [53]. The solid curve is based on the static approximation (SA) to the dynamical CPA. The closed circles are the results of the Quantum Monte-Carlo method (QMC). The energy unit is chosen to be W = 1. Below TN the antiferromagnetic state (AF) is stabilized. Above TN there are the paramagnetic metal (PM) and the paramagnetic insulator (PI) regimes. The open squares indicate a crossover line between the two states

Fig. 6.3 Typical magnetic phase diagram on the U –n plane. The result is calculated with use of the Gutzwiller approximation for the Hubbard model on the hyper-cubic lattice in infinite dimension [98]

magnetic state (P) with increasing electron number from n = 1. When the Coulomb interaction is large enough, the transition from the AF to the ferromagnetic state (F) occurs with increasing electron number. It should be noted that the phase diagram

6.2 Generalized Static Susceptibility and Antiferromagnetism

157

is symmetric around the n = 1 axis when the electron-hole particle symmetry is present. In the above theoretical treatment of the AF state, we assumed that there is no electron hopping between two sublattices. Furthermore we used a symmetric property of the DOS in the nonmagnetic state. These assumptions are not satisfied in general. For example, in the case of the first-kind AF state on the fcc lattice as shown in Fig. 1.10(c), the transfer integrals between atoms belonging to the same sublattice are not negligible because of the same interatomic distance. Moreover the DOS is not symmetric in the case of the fcc lattice. In this case, the energy gap on the Fermi level is not expected to appear, and the kinetic energy gain due to alternative exchange splitting is not so effective; we do not expect the appearance of the gap-type AF ordered state. The antiferromagnetic metals which are not accompanied by the band gap are referred as the band-type antiferromagnets. The critical Coulomb interaction for the appearance of the AF state is expected to be finite even at half filling in this case.

6.2 Generalized Static Susceptibility and Antiferromagnetism An alternative way to understand the stability of magnetic structure is to analyze the generalized susceptibility. Let us assume that a metal is in the paramagnetic state when there is no external field. We then consider a linear polarization due to sitedependent magnetic fields {hi }. The linear polarization of magnetic moment on site i is given by  χij hj . (6.28) mi  = j

The above expression defines the nonlocal static susceptibility χij . The susceptibilities in the real space give us the magnetic couplings between the local magnetic moments in metals, and are useful for understanding the complex magnetic structure. In order to obtain the susceptibility in the Hartree–Fock approximation, we add the Zeeman energy to the atomic level of the Hamiltonian (6.1) as εiσ = ε0 + U ni /2 − (U mi /2 + hj )σ , and express the Green function (6.5) by means of the locator matrix Lij σ such that (L−1 σ )ij = (z − ε0 + μ − U ni /2 + (U mi /2 + hi )σ )δij as   Giiσ (z) = (Lσ − t)−1 ii .

(6.29)

Here t denotes the transfer integral matrix tij . Note that the energy is measured from the Fermi level here. The linear change of the locator due to spin polarization is given by δL−1 iσ =  −1 (U j χij hj /2 + hi )σ , and that of the Green function is given by −[Gσ δLσ Gσ ]ii . Substituting the Green function and the linear change of magnetic moment (6.28)

158

6 Antiferromagnetism and Spin Density Waves

into the self-consistent equation (6.3), we find the equation for nonlocal susceptibility as follows. χij = χij0 +

1 U χil0 χlj . 2

(6.30)

l

Here χij0 is the susceptibility for the noninteracting system given by  χij0

=

dω f (ω)

(−)  Im Gij σ (z)Gj iσ (z). π σ

(6.31)

The Green function for the noninteracting system is given by Gij σ =

1 1 e−ik·(Ri −Rj ) . L z − ε˜ k

(6.32)

k

Here ε˜ k = ε0 − μ + U n/2 + εk . The magnetic coupling in metals and alloys are generally long range, so that their Fourier representations are often useful. The Fourier transform of the linear response (6.28) is given by m(q) = χ(q)h(q). (6.33)   Here m(q) (h(q)) is defined by mi  = q m(q) exp(iq ·R i ) (hi = q h(q) exp(iq · R i )). The susceptibility χ(q) for the wave vector q is defined by χij =

1 χ(q) e iq·(Ri −Rj ) . L q

(6.34)

Substituting (6.34) into (6.30), we find the Fourier representation of the Hartree– Fock susceptibility. χ(q) =

χ0 (q) 1 − 12 U χ0 (q)

.

(6.35)

The susceptibility for the noninteracting system is obtained from (6.31) as χ0 (q) =

2  f (˜εk+q ) − f (˜εk ) . L εk − εk+q

(6.36)

k

The result agrees with χ0zz (q, ω = 0) = χ0−+ (q, ω = 0)/2 of (4.101) as it should be. At half-filling (μ = ε0 + U n/2) and for Q such that εk+Q = −εk (e.g., Q = (π/a)(±1, ±1, ±1) for the sc lattice and Q = ±(2π/a)(0, 0, 1) for the bcc lattice), we obtain  tanh(βε/2) χ0 (Q) = dε ρ(ε) . (6.37) ε

6.2 Generalized Static Susceptibility and Antiferromagnetism

159

Here ρ(ε) is the noninteracting DOS per atom per spin. The r.h.s. has the same form as that of (6.22). Making the approximation (6.23), we find χ0 (Q) = 2ρ(0) ln

βγ W . π

(6.38)

The above expression shows that the susceptibility for the noninteracting system diverges as T goes to zero, thus according to (6.35) the susceptibility χ(Q) also diverges at a certain temperature TN with decreasing temperature. This means that the antiferromagnetic state with a sublattice magnetization m(Q) should be stabilized below TN . The Néel temperature TN in the Hartree–Fock approximation is obtained from the condition 1 = U χ0 (Q)/2. Using (6.38), we obtain the following gap-type Néel temperature. TN =

1 γ W − ρ(0)U W − 1 e ≈ 1.13 e ρ(0)U . π 2

(6.39)

The result agrees with (6.24). The generalized susceptibility χ(q) is extended to the multi-band case. Let us adopt the nearly orthogonal basis set {χiL } for tight-binding linear muffin-tin orbitals (TB-LMTO) (see (2.154)), and assume a system with one atom per unit cell for simplicity. The eigen state ψnk (r) to the Hamiltonian (2.167) is given by ψnk (r) =



χiL (r)iL|nk,

(6.40)

1 iL|nk = uˆ Ln (k) √ e−ik·Ri . N

(6.41)

il

Here i (L) denotes site (orbital), and N is the number of unit cells. uˆ Ln (k) is the eigen vector for a given momentum k. When we insert infinitesimal magnetic field hj L on each site j and orbital L into the TB-LMTO Hamiltonian (2.167), we have a linear polarization of the magnetic moment on site i and orbital L as miL  =

 j L

(0)

χiLj L hj L .

(6.42)

(0)

Here χiLj L is the nonlocal susceptibility for the noninteracting system. In order to obtain the nonlocal susceptibility as shown in (6.28), we have to apply the magnetic field hj L = hj . We have then (0)

χij =

 LL

(0)

χiLj L .

(6.43)

160

6 Antiferromagnetism and Spin Density Waves

Taking the same steps as in the single band model, we find the unenhanced susceptibility at zero temperature as follows. 



| L uˆ ∗Ln (k)uˆ Ln (k + q)|2 2   χ0 (q) = f ε˜ n (k) − f ε˜ n (k + q) . N  εn (k + q) − εn (k) n

nk

(6.44) Here ε˜ n (k) = εn (k) − μ. The full susceptibility χ(q) is enhanced by the exchange potential (−I mi σ/2) as follows. χ(q) =

χ0 (q) 1 − 12 I χ0 (q)

.

(6.45)

Here I is the Stoner parameter defined by (2.176). The ferromagnetic structure (F) may be realized when 2 χ0 (0) = 2ρ(0) > , I

(6.46)

where ρ(0) is the total density of states per atom per spin at the Fermi level in the nonmagnetic state. This is the Stoner criterion (2.181) discussed in Sect. 2.3.4. The antiferromagnetic state (AF) is realized when 2 χ0 (Q) > . I

(6.47)

Here Q = (0, 0, 1)2π/a in the case of the bcc and fcc structures, a being the lattice constant. χ0 (0) (χ0 (Q)) is called the uniform (staggered) susceptibility. Whether the AF or F is stable might be determined by comparing the uniform susceptibility χ0 (0) with the staggered χ0 (Q). Figure 6.4 shows the susceptibilities χ0 (0) and χ0 (Q) as a function of the conduction electron number (n) [99, 100]. In the case of the bcc Cr (n = 6), we find a huge χ0 (Q), indicating the appearance of a gap-type AF. The uniform susceptibilities of bcc Fe (n = 8) and fcc Ni (n = 10) show a large value, which indicates the ferromagnetism in these systems in agreement with the experimental fact. The ferromagnetism of fcc Co is not clear from the figure because χ0 (0) of fcc Co is not so high. Note that Co shows the hcp structure at T = 0, though it shows the fcc structure above 700 K. It is interesting that χ0 (Q) > χ0 (0) in the case of the fcc Fe, while χ0 (0)  χ0 (Q) in the case of bcc Fe. The result suggests the AF in fcc Fe. Experimentally, the pure bcc Cr shows the spin density waves (SDW) structure with the wave vector Q = (0, 0, 0.95)2π/a [101], though Cr with a few atomic percent of Mn does show the AF structure (see Fig. 6.1). The fcc Fe is reported to show the SDW with Q = (1, 0, 0.15)2π/a [102], though it shows the first-kind AF when Mn is added by several percent [103]. A large enhancement of χ0 (Q) at Cr in Fig. 6.4 is considered to be a Fermisurface nesting effect. According to the expression (6.44), the susceptibility χ0 (Q)

6.2 Generalized Static Susceptibility and Antiferromagnetism

161

Fig. 6.4 Non-interacting spin susceptibilities χ0 (0) (solid curves) and χ0 (Q) (dashed curves) as a function of the conduction electron number for bcc transition metals (left) and fcc transition metals (right) [99, 100]. The wave vector is given by Q = (0, 0, 1)2π/a for both structures, a being the lattice constant

is enhanced when there are many electron k points in the vicinity of Fermi surface and at the same time the k + Q hole points are also in the vicinity of the Fermi surface, because the denominator εn (k + Q) − εn (k) almost vanishes for such k points. Such an enhancement is expected when there are a large electron Fermi surface and a hole Fermi surface with the same shape which are distant from each other by Q. In this case the Fermi surfaces are called ‘being nested’, and the enhancement of the susceptibility due to the nesting mechanism is called the Fermi surface nesting effect. Note that the divergence of the susceptibility in (6.38) at T = 0 for the single-band Hubbard model at half-filling is also due to the complete nesting because εkF +Q = εkF = 0 for the Fermi wave vector kF . Figure 6.5 shows a schematic Fermi surface of Cr on the (001) intersection of the Brillouin zone [104]. According to the electronic structure calculations, there is an electron Fermi surface of a nearly octahedral shape around the point (0, 0, 0), and there is a hole Fermi surface around the H point (1, 0, 0)2π/a which is also octahedral in shape, though slightly larger than the former around the point. These Fermi surfaces are approximately nested with the wave vector Q = (0, 0, 1)2π/a. This is the reason why χ0 (Q) is high for Cr in Fig. 6.4. However, we may expect that 50 percent of complete nesting of the Fermi surface using the wave vector Q = (0, 0, 0.95)2π/a may be more favorable for the enhancement of χ0 (Q), because the hole Fermi surface is slightly larger than the electron one as shown in Fig. 6.5. This is why the SDW state with Q = (0, 0, 0.95)2π/a, instead of the AF structure, is realized in Cr.

162

6 Antiferromagnetism and Spin Density Waves

Fig. 6.5 Schematic (001) intersection of chromium Fermi surface

When we add Mn atoms to Cr, the average electron number is increased, so that the octahedral electron Fermi surface grows up in size, while the hole Fermi surface around H point becomes smaller. This implies that the nesting wave vector increases in magnitude, and the AF state is realized as found in the phase diagram, Fig. 6.1. On the other hand, if the neighboring element V is added to Cr, the average electron number decreases, so that the electron Fermi surface shrinks and the hole Fermi surface expands. As the result the enhancement of χ0 (Q) due to the nesting is weakened and the paramagnetic state is realized in agreement with the experimental data (see Fig. 6.1). In the case of the first-kind AF structure in fcc Mn, the Fermi surface nesting is not found. For the fcc structure, neither the gap type of AF as mentioned in Sect. 6.1 nor equivalently the nesting type of AF as have been explained in this section are expected because the hole-particle symmetry of the band is not expected even at half-filling. One has to consider the stability based on the total energy calculation for such a system. Instead of uniform polarization in the ferromagnetism, the local spin polarization as expressed by (6.3), thus the local densities of states (6.4) are essential there. When the Coulomb energy gain due to the local polarization overcomes the band energy loss, the AF is stabilized [105]. The antiferromagnets based on this mechanism are called the band-type AF.

6.3 Molecular Dynamics Theory for Complex Magnetic Structures As we have seen in Fig. 6.3, the ferromagnetic state (F) is stabilized in the region of few electrons or nearly filled shell, while the antiferromagnetic state (AF) is stabilized in the half-filled region. Near the boundary between the F and the AF regions, we can expect competition between the ferro- and the antiferro-magnetic interactions, and thus the appearance of various complex magnetic structures. It is not easy however to find intuitively the complex magnetic structures which yield

6.3 Molecular Dynamics Theory for Complex Magnetic Structures

163

the global minimum of the free energy. In this section, we present a molecular dynamics method which automatically determines the complex magnetic structure [108, 109]. We adopt the tight-binding LMTO Hamiltonian (3.228) and the functional integral method. The free energy F is then given by (3.245), and the time-dependent Hamiltonian H (τ, ξ (τ ), −iζ (τ )) in the free energy is given by (3.246). Since we are considering the complex magnetic structure due to the competition between the long-range ferro- and antiferromagnetic couplings, we should take into account intersite correlations. Instead, we adopt in this section the static approximation which neglects the dynamical spin and charge fluctuations. As shown in Sect. 3.2, we approximate the time-dependent Hamiltonian H (τ, ξ (τ ), −iζ (τ )) given by (3.246) with the static one H (τ, ξ , −iζ ), where ξ im (ζim ) are the static exchange (charge) β β field variables defined by ξ im = 0 ξ im (τ ) dτ (ζim = 0 ζim (τ ) dτ ). Then the partition function Z 0 (ξ (τ ), ζ (τ )) is given by

Z 0 (ξ , ζ ) = Tr e−βH (0,ξ ,−iζ ) .

(6.48)

The remaining Gaussian integrals in the free energy can be performed in the same way as in Sect. 3.2 (see (3.57) and (3.58)). Applying the saddle point approximation to the charge fields, we obtain the free energy as follows. F = −β

−1

ln

   β 2l+1 detB α  iα

(4π)2l+1

dξimα e−βE(ξ ) .

(6.49)

m

Here and in the following l stands for l = 2 assuming the 3d electron system. The effective potential in the static approximation is given by

E(ξ ) = −β −1 ln tr e−βH (ξ )    1  α n˜ ilm (ξ )Aimm n˜ ilm (ξ ) − ξimα Bimm ξim α . − 4  α

(6.50)

i mm

The Hamiltonian H (ξ ) is given by   1 0 H (ξ ) = εiL −μ+ Amm n˜ ilm (ξ )δld niL 2  iL

m

−

1  α Bimm ξim α mαilm δld , 2 α 

(6.51)

m

and n˜ ilm (ξ ) denotes the charge density on site i and orbital lm in the static approximation when the exchange fields {ξ im } are given. μ denotes the chemical potential. α The Coulomb and exchange interaction matrices Amm and Bimm  are defined by (3.247), (3.248), and (3.249), respectively.

164

6 Antiferromagnetism and Spin Density Waves

In order to reduce the number of variables, we replace the orbital-dependent charge densities {n˜ ilm (ξ )} (exchange field ξimα ) in the free energy with an averaged n˜ il (ξ )/(2l + 1) (ξiα /(2l + 1)). Here n˜ il (ξ ) and ξiα denote the d-electron charge density  on site i and the total  exchange field on the same site, respectively; n˜ il (ξ ) = m n˜ ilm (ξ ) and ξiα = m ξimα . The simplified free energy is then expressed as  



β J˜α dξiα e−βE(ξ ) , F = −β ln 4π iα   

1 ˜ 2 , E(ξ ) = −β −1 ln tr e−βH (ξ ) − J˜α ξiα U n˜ il (ξ )2 − 4 α i   

1 ˜ 1 ˜ 0 H (ξ ) = εiL − μ niL + Jα ξiα mαilm δld U n˜ il (ξ )nilm − 2 2 α iL  † tiLj L aiLσ aj L σ . + −1

(6.52)

(6.53)

(6.54)

iLj L σ

Here U˜ = U0 /(2l + 1) + [1 − 1/(2l + 1)](2U1 − J ) and J˜α = U0 δαz /(2l + 1) + [1 − 1/(2l + 1)]J . The free energy (6.52) does not satisfy the rotational invariance in spin space because of the anisotropic exchange interaction J˜α inherent in the Hartree–Fock type static approximation. In order to avoid the problem within the static approximation, we introduce the locally rotated coordinates at each site. The interaction term of the Hamiltonian in the locally rotated coordinates has the same form as the original (3.230) due to the rotational invariance. But  the operators have been replaced by those on the rotated coordinates; n˘ ilm = σ n˘ ilmσ and s˘ ilm =  † † ˘ ilmα (σ /2)αγ a˘ ilmγ . Here a˘ iLα and a˘ iLα are the creation and annihilation opαγ a  † † erators on the rotated coordinates, which are given by a˘ iLα = γ aiLγ Dγ α (Ri ) and  ∗ a˘ iLα = γ aiLγ Dγ α (Ri ) with use of the rotation matrix Dαγ (Ri ) for a spin on site i. Here Ri denotes a rotation of the z axis. We apply the functional integral technique to the Hamiltonian on locally rotated coordinates. Taking the same steps as before we obtain the counterpart of the free energy (6.52). We then neglect the transverse static spin fluctuations on the rotated coordinates and take the average of the free energy over the z direction of the locally rotated coordinates, so that we obtain the free energy as follows.    β J˜ F = −β −1 ln dξi dei e−βE(ξ ) , (6.55) 4π i

1 

E(ξ ) = −β −1 ln tr e−βH (ξ ) − U˜ n˜ il (ξ )2 − J˜ξ 2i , 4 i

(6.56)

6.3 Molecular Dynamics Theory for Complex Magnetic Structures

H (ξ ) =

 iL

+

 0 −μ+ εiL



iLj L σ

165

 1 ˜ 1 U n˜ il (ξ )nil − J˜ξ i · mil δld 2 2

† tiLj L aiLσ aj L σ .

(6.57)

Here J˜ = U0 /(2l + 1) + [1 − 1/(2l + 1)]J , ei is the unit vector on site i showing the direction of rotated z axis, and ξi is the z component of the exchange fields on the rotated coordinates. Furthermore ξ i = ξi ei and dei = (4π)−1 sin θi dθi dφi . θi and φi denote the zenith and azimuth angles of vector ei . Since n˜ ilm (ξ ) is a saddle point value of the charge field ζilm (see (3.66)), it is given by tr(nilm e−βH (ξ ) ) . (6.58) tr(e−βH (ξ ) ) The local charge and magnetic moment are obtained by taking the derivative of F 0 and the local magnetic field h acting on site i. with respect to the atomic level εiL i   n˜ i  = n˜ iL (ξ ) , (6.59) n˜ ilm (ξ ) = nilm 0 =

 mi  =

L

  4 ξi . 1+ β J˜ξ 2

(6.60)

i

Here the average ∼ at the r.h.s. of the equations is defined by   dξ i (∼) e−βΨ (ξ ) ∼ =

i  

dξ i e−βΨ (ξ )

i

Ψ (ξ ) = E(ξ ) + 2β −1



ln ξi .

,

(6.61)

(6.62)

i

Note that we have adopted the spherical coordinates in the above average, and thus dξ i = ξi2 sin θi dξi dθi dφi . An alternative way to recover the rotational invariance of the free energy within the static approximation is to take the limit l → ∞ (see (6.52)). We obtain in this case J˜α = J and the free energy (6.55) in which dξi dei has been replaced by dξ i . The magnetic moment mi  is given by (6.60) in which the prefactor (1 + 4/β J˜ξi2 ) has been replaced by 1, and the potential Ψ (ξ ) (i.e., (6.62)) reduces to E(ξ ). In any case, the Coulomb and exchange energies, U˜ and J˜, have to be regarded as effective ones since we adopted the static approximation which neglects the electron correlations at low temperatures. Equation (6.60) indicates that the thermal average of local magnetic moment (LM) is given by a semiclassical average with respect to the potential energy Ψ (ξ ). In this case, we can apply the isothermal molecular-dynamics method (MD) [106, 107] in order to treat a large number of atomic magnetic moments {mi }.

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6 Antiferromagnetism and Spin Density Waves

In the MD method we express the magnetic moment (6.60) by means of the time average assuming the ergodicity of the system as follows.    4 1 t0 1+ ξ i (t) dt. mi  = lim (6.63) t0 →∞ t0 0 β J˜ξi2 (t) The dynamics of the ‘time-dependent magnetic moment’ {ξ i (t)} which yields the thermal average (6.60) are given by the following equations of motion (see Appendix F) [108]. ξ˙iα =

piα , μLM

∂Ψ (ξ ) − ηα · piα , ∂ξiα   2 1  piα − NT . η˙ α = Q μLM

p˙iα = −

(6.64) (6.65) (6.66)

i

Here piα is a fictitious momentum conjugate to the exchange field variable ξiα . μLM is an effective mass for the LM on site i. The first term at the right-hand-side of (6.65) is a magnetic force, and the second term is the friction force which keeps temperature T constant according to (6.66). Q in (6.66) denotes a constant parameter, and N is the number of atoms in the system. Note that we have introduced the anisotropic friction variables ηα (α = x, y, z) to guarantee the ergodicity of the system even if N = 1. The magnetic force in (6.65) is obtained from (6.56) and (6.62) as −

2T ξiα ∂Ψ (ξ ) 1 ˜ = J miα 0 − ξiα − . ∂ξiα 2 ξi2

(6.67)

Here miα 0 is the average magnetic moment with respect to the Hamiltonian (6.57) in the random exchange fields. It is given by the Green function G(z) for the Hamiltonian as 

(−)  miα 0 = dω f (ω) Im σα G(ω + iδ) ilmσ ilmσ , (6.68) π mσ Gilmσ j l  m σ  (z) =



−1  z − H (ξ ) . ilmσ j l  m σ 

(6.69)

Here l = 2, and H (ξ ) is the one-electron Hamiltonian matrix of (6.57). The Green functions coupled with the Pauli spin matrices in (6.68) are expressed as follows with use of the new basis functions which diagonalize the Pauli spin matrices σα (α = x, y, z).  (σx G)iLσ iLσ = GiL1iL1 − GiL2iL2 , (6.70) σ

6.3 Molecular Dynamics Theory for Complex Magnetic Structures

167

Fig. 6.6 The MD unit cell embedded by the site-dependent effective medium ΣiLσ σ  and site-independent effective medium ΣLσ

 (σy G)iLσ iLσ = GiL3iL3 − GiL4iL4 ,

(6.71)

σ

 (σz G)iLσ iLσ = GiL↑iL↑ − GiL↓iL↓ .

(6.72)

σ

Here the local basis functions at the r.h.s. are defined by

1 |iL1 = √ |iL ↑ + |iL ↓ , 2

1 |iL2 = √ |iL ↑ − |iL ↓ , 2

1 |iL3 = √ |iL ↑ + i|iL ↓ , 2

1 |iL4 = √ |iL ↑ − i|iL ↓ . 2

(6.73) (6.74) (6.75) (6.76)

It should be noted that the exchange fields {ξ i } randomly change in space and time in the MD method, so that the system does not satisfy in general the translational symmetry at each time step in the MD. Therefore, the recursion method is used for the calculation of the Green functions in (6.70), (6.71), and (6.72) (see Appendix G). In numerical calculations, we have to consider the MD unit cell with the finite number of atoms N . The ‘magnetic moment’ ξ i is centered in the MD unit cell. In order to simulate the system with N = ∞, the MD unit cell is usually embedded in a site-dependent effective medium ΣiLσ σ  . It is further surrounded by a siteindependent medium as shown in Fig. 6.6. They are determined self-consistently by the CPA equations (3.254) in which the dynamical correction has been omitted and iωl has been replaced by z. The fcc transition metals are expected to form complex magnetic structures around the d electron numbers nd between 6.0 and 7.0 due to competing interactions. Some numerical results of the model calculations are presented in Figs. 6.7,

168

6 Antiferromagnetism and Spin Density Waves

Fig. 6.7 Magnetic structures of the fcc transition metals at 50 K obtained by the MD with 108 atoms per unit cell: case of the d electron number n = 6.2 [109]

Fig. 6.8 Magnetic structures of the fcc transition metals obtained by the MD with 108 atoms per unit cell: case of the d electron number n = 6.4 [109]

6.8, and 6.9 [108, 109]. The Slater–Koster d band model [31] and N = 108 atoms per MD unit cell (3 × 3 × 3 fcc lattice) are adopted there. The band width and effective exchange energy parameter were fixed to be the values of γ -Mn: W = 0.443 Ry and J˜ = 0.060 Ry. Starting from a random configuration of LM’s {ξ i (0)}, we solve (6.64)–(6.66) and obtain the magnetic moments mi  by taking the time average in the equilibrium state. The calculated magnetic structure for nd = 6.2 shows the AF structure of the first kind in accordance with the magnetic structure of γ -Mn as shown in Fig. 6.7. The calculated magnetic moment |mi | = 2.5 μB is also consistent with the experimental value 2.3 μB and the theoretical one 2.32 μB [105]. The MD method yields the Néel temperature TN = 510 K for γ -Mn [110] which is in good agreement with the experimental result TN = 500 K. When the d electron number nd is increased, the AF structure changes to the helical structures due to competition between the ferro- and antiferro-magnetic interactions. For n = 6.4, the LM’s in an antiferromagnetic plane rotate by 240◦ with a translation by the lattice constant a along the axis perpendicular to the

6.4 Phenomenological Theory of Magnetic Structure

169

Fig. 6.9 Magnetic structures of the fcc transition metals obtained by the MD with 108 atoms per unit cell: case of the d electron number n = 6.6 [109]

plane as shown in Figs. 6.8. The wave vector of the helical structure is given by Q = (0, 1/3, 1)2π/a and |mi | = 2.0 μB . A further increase in d electron number reduces the average magnetic moments and leads to the helical structure with amplitude modulations as shown in Fig. 6.9; the wave vector Q is the same as in n = 6.4, but the magnitudes of LM |mi | are spatially modulated from 0.75 μB to 1.30 μB . The modulated structure is characteristic of itinerant electron systems because such a modulation should be suppressed by a large energy loss of Coulomb interactions in the insulator system. It is stabilized by the energy gain due to the break of a frustrated magnetic structure with equal amplitudes of LM’s. The MD approach is useful for theoretical study of the magnetic structure in competing magnetic interaction system. The size of the MD unit cell is however limited to be finite and thus the SDW with wave length larger than the lattice constant of the MD unit cell are not described.

6.4 Phenomenological Theory of Magnetic Structure Competing magnetic interactions often form the complex magnetic structures whose unit cells are huge or which are described by incommensurate wave vectors. Microscopic theories of magnetic structure are often faced with the difficulty in description for such systems because the system size which one can treat is limited to rather small number (104 ). Furthermore the accuracy of the microscopic theories is often not enough to describe the stability of magnetic structures when the energy difference between structures is too small. In such cases we have to rely on the phenomenological theory to understand the structure. We present in this section the Ginzburg–Landau phenomenological theory, and discuss possible multiple spin density waves (MSDW) in γ -Fe [111, 112]. The theory is also useful for finding general properties of the magnetic structure determined by crystal symmetry. Let us consider a system with a magnetic atom per unit cell and assume that it shows a cubic crystal symmetry for simplicity. Assuming small size of magnetic

170

6 Antiferromagnetism and Spin Density Waves

moments, we expand the free energy F with respect to the local magnetic moments. Because of the time reversal symmetry in the absence of the external magnetic field, the free energy consists of even-order terms of the magnetic moments. We consider here the terms up to the fourth order. Because of cubic symmetry the free energy should be invariant with respect to the symmetry operations for magnetic moments: rotation mRl → R(mR l ), inversion mRl → m−R l , and translations mR l → mR l +T . Here R l is the position vector of site l, and mR l ≡ ml is the thermal average of the local magnetic moment on site l. R denotes either the rotation C4 [100] or C3 [111], and T denotes an arbitrary lattice translation vector. The free energy of the system per lattice site is then expressed as follows. F=

1  

1 A l, l ml · ml  + 4 2 N N  l, l

 

B l, l  , l  , l  {ml · ml  }{ml  · ml  } l, l  , l  , l 



+ C l, l  , l  , l  (mly ml  y ml  z ml  z + mlz ml  z ml  y ml  y + mlz ml  z ml  x ml  x  + mlx ml  x ml  z ml  z + mlx ml  x ml  y ml  y + mly ml  y ml  x ml  x ) . (6.77) Here N is the number of lattice sites. A(l, l  ), B(l, l  , l  , l  ), and C(l, l  , l  , l  ) are expansion coefficients. The second-order terms and the fourth-order terms with B(l, l  , l  , l  ) in the free energy (6.77) are isotropic since they are expressed in terms of the scalar products of magnetic moment vectors. On the other hand, the fourth-order terms with coefficients C(l, l  , l  , l  ) are anisotropic. Such an anisotropy is caused by the spin–orbit interactions. Here we restrict ourselves to the transition metals where the spin–orbit coupling effects are negligibly small, and neglect the anisotropic terms; C(l, l  , l  , l  ) = 0. It is convenient to use the Fourier representation of the local magnetic moment in order to describe the spin density wave (SDW). ml =

EBZ 

m(q)eiq·R l .

(6.78)

q

 means a summation with respect to q over the extended first Brillouin Here EBZ q zone (EBZ), which is defined to include all zone boundary points. This form has the merit that one can argue the structures in both commensurate and incommensurate cases on the same footing. The Fourier representation of the isotropic free energy is then given by F=

EBZ 

 2   2   A(q) m(q) + δq,K/2 m (q)

q

+





K =0

! ! B q, q  , q  , q  m(q) · m q  m q  · m q  .

K q+q  +q  +q  =K

(6.79)

6.4 Phenomenological Theory of Magnetic Structure

171

Fig. 6.10 Phenomenological Landau free energy F of ferromagnet as a function of the magnetization M below and above the Curie temperature (TC )

Here K is the reciprocal lattice vector. A(q) and B(q, q  , q  , q  ) are coefficients of the free energy expansion in the Fourier representation. In the ferromagnetic state, only m ≡ |m(q = 0)| component remains. We have then the free energy, FF = Am2 + Bm4 − mh.

(6.80)

Here A ≡ A(0), B ≡ B(0, 0, 0, 0), and the term of uniform magnetic field h is added. The free energies below and above the Curie temperature are depicted for h = 0 in Fig. 6.10. Equilibrium magnetization is obtained from the stationary condition.

m 2A + 4Bm3 = h.

(6.81)

Above the Curie temperature TC and for small h, we have m = h/2A. Thus A = 1/2χ , χ being the paramagnetic susceptibility. At TC the susceptibility χ diverges. When we expand A as A = (T − TC )/2C above TC , we obtain the Curie–Weiss law χ = C/(T − TC ). Below TC and √ for h = 0, (6.81) is identical with (3.18). We have the magnetization (3.21); m = −A/2B under the condition −A/2B > 0. The stability condition for the ferromagnetic solution is given by ∂ 2 FF /∂ 2 m < 0 at the equilibrium m, and yields A < 0. From the conditions −A/2B > 0 and A√< 0, we obtain B > 0. Expanding A with respect to T − TC below TC , we find m ∝ TC − T near TC . The free energy in the equilibrium state is given by FF = −A2 /4B. This is the phenomenological theory of ferromagnetism known as the Landau theory.

172

6 Antiferromagnetism and Spin Density Waves

The same argument is applicable to the antiferromagnetic structure (AF) which is given by

ml = m(Q)k eiQ·R l + e−iQ·R l .

(6.82)

Here Q is a wave vector leading to the AF structure (see Sect. 1.7). k is a unit vector along the z axis. The free energy of the AF is given by FAF = A(Q)m(Q)2 + B(Q)m(Q)4 − m(Q)h(Q).

(6.83)

Here A(Q) and B(Q) are coefficients, and h(Q) is the staggard field. Minimizing the free energy, the sublattice magnetization is obtained as m(Q) = √ −A(Q)/2B(Q) when A(Q) < 0 and B(Q) > 0. The free energy in the equilibrium state is given by FAF = −A(Q)2 /4B(Q). It is also worthwhile to mention the long-wavelength limit of the free energy (6.79). There only the q components around |q| = 0 are important there, so that one can replace B(q, q  , q  , q  ) with a constant γ /4(= B(0, 0, 0, 0)). F=

EBZ  q

2 1 1  m(q) + γ 2χ(q) 4



! ! m(q) · m q  m q  · m q  .

q+q  +q  +q  =0

(6.84) Here χ(q) is the generalized susceptibility defined by (6.33): m(q) = χ(q)h(q). The form (6.84) is used in the phenomenological spin-fluctuation theory for the weak ferromagnets [113].

6.4.1 Multiple SDW with Commensurate Wave Vectors In the itinerant electron system with high crystalline symmetry, the multiple spin density wave (MSDW) structure becomes possible. We consider in this subsection the commensurate multiple MSDW structures (see Fig. 1.15). We consider the fcc lattice here as an example. The magnetic moments for the MSDW are expressed by ml =

3    ˆ n ) ei Qˆ n ·R l + m(Q ˆ n ) e−i Qˆ n ·R l . m(Q

(6.85)

n=1

ˆ 1 = (1, 0, 0)(2π/a), Q ˆ 2 = (0, 1, 0)(2π/a), Here the wave vectors are given by Q ˆ and Q3 = (0, 0, 1)(2π/a). Each wave vector forms the antiferromagnetic strucˆ 2 ), and m(Q ˆ 3 ) are real ˆ 1 ), m(Q ture of the first kind (AF-I) on the fcc lattice. m(Q ˆ 3 ) = m(Q ˆ 3 ) · m(Q ˆ 1) = ˆ 2 ) · m(Q and assumed to be orthogonal to each other: m(Q ˆ 2 ) = 0. ˆ 1 ) · m(Q m(Q

6.4 Phenomenological Theory of Magnetic Structure

173

ˆ i } in the free energy (6.79), we obtain Taking the terms related with {Q F=

3        ˆ i )4 ˆ i )2 + (B1Q + B˜ 2Q )m(Q A˜ Q m(Q i=1

+

(2,3)(3,1)(1,2) 

    ˆ i )2 m(Q ˆ j )2 . B˜ 1QQ m(Q

(6.86)

(i,j )

The coefficients A˜ Q , B1Q , B˜ 2Q , and B˜ 1QQ are expressed in terms of linear combinations of the coefficients {A(q)} and {B(q, q  , q  , q  )}, where q, q  , q  , and q  ˆ 1 , ±Q ˆ 2 , and ±Q ˆ 3. are chosen to be one of ±Q ˆ 2 ) = m(Q ˆ 3 ) = 0) is ˆ 1 ) = 0, m(Q As mentioned before, the AF-I structure (m(Q stable in the region A˜ Q < 0,

(6.87)

B1Q + B˜ 2Q > 0.

(6.88)

We obtain the magnetic moment    m(Q1 ) = −

1/2

A˜ Q 2(B1Q + B˜ 2Q )

,

(6.89)

as well as the free energy F1Qˆ = −

A˜ 2Q 4(B1Q + B˜ 2Q )

.

(6.90)

ˆ state is characterized by m(Q ˆ 1 ), m(Q ˆ 2 ) = 0 and The double Q MSDW (2Q) ˆ 3 ) = 0 (see Fig. 1.14). The stationary condition of the free energy yields the m(Q equations for magnetic moments as     ˆ 1 )2 + B˜ 1QQ m(Q ˆ 2 )2 = 0, A˜ Q + 2(B1Q + B˜ 2Q )m(Q     ˆ 2 )2 + B˜ 1QQ m(Q ˆ 1 )2 = 0. A˜ Q + 2(B1Q + B˜ 2Q )m(Q

(6.91) (6.92)

When 2 D2Qˆ ≡ 4(B1Q + B˜ 2Q )2 − B˜ 1QQ = 0,

(6.93)

equations (6.91) and (6.92) are solved as      m(Q ˆ 2 ) = − ˆ 1 ) = m(Q

1/2 A˜ Q , 2(B1Q + B˜ 2Q ) + B˜ 1QQ

(6.94)

174

6 Antiferromagnetism and Spin Density Waves

under the condition −

A˜ Q > 0. 2(B1Q + B˜ 2Q ) + B˜ 1QQ

(6.95)

The stability condition to the solution (6.94) is given by 2 2     ∂ 2F 1  ˆ j )2 > 0. ˆ i )2 δ m(Q δ F= δ m(Q ˆ i )|2 }∂{|m(Q ˆ j )|2 } 2 ∂{|m(Q 2

(6.96)

i=1 j =1

It is equivalent to f11 > 0,

  f11   f21

 f12  > 0, f22 

(6.97)

ˆ i )|2 }∂{|m(Q ˆ j )|2 } (i, j = 1, 2). The where fij is defined by fij ≡ ∂ 2 F /∂{|m(Q condition (6.97) yields the inequalities, 2(B1Q + B˜ 2Q ) > 0,

(6.98)

2 > 0. 4(B1Q + B˜ 2Q )2 − B˜ 1QQ

(6.99)

Conditions (6.93), (6.95), (6.98), and (6.99) reduce to A˜ Q < 0,

(6.100)

|B˜ 1QQ | . B1Q + B˜ 2Q > 2

(6.101)

ˆ structure. Inequalities (6.100) and (6.101) yield the stability condition for the 2Q ˆ The equilibrium free energy is obtained by substituting (6.94) and m(Q3 ) = 0 into (6.86): F2Qˆ = −

A˜ 2Q 2(B1Q + B˜ 2Q ) + B˜ 1QQ

.

(6.102)

ˆ In the same way, we obtain magnetic moments of the triple Q MSDW (3Q) 2 ˆ ˆ structure by minimizing free energy (6.86) with respect to |m(Q1 )| , |m(Q2 )|2 , ˆ 3 )|2 . and |m(Q        m(Q ˆ 2 ) = m(Q ˆ 3 ) = − ˆ 1 ) = m(Q

1/2 A˜ Q , 2(B1Q + B˜ 2Q + B˜ 1QQ )

(6.103)

under the condition −

A˜ Q > 0. 2(B1Q + B˜ 2Q + B˜ 1QQ )

(6.104)

6.4 Phenomenological Theory of Magnetic Structure

175

ˆ structure and inequality (6.104) The thermodynamical stability condition for the 3Q lead to the following condition: A˜ Q < 0,

(6.105)

B˜ 1QQ B1Q + B˜ 2Q > for B˜ 1QQ > 0, 2 B1Q + B˜ 2Q > −B˜ 1QQ for B˜ 1QQ < 0.

(6.106) (6.107)

The equilibrium free energy is obtained by substituting (6.103) into (6.86): F3Qˆ = −

3A˜ 2Q 4(B1Q + B˜ 2Q + B˜ 1QQ )

.

(6.108)

It is characteristic of the itinerant electron system that the amplitudes of the magnetic moments are variable. The amplitude M per site is given by EBZ   1  M ≡ ml · ml = m(q) · m q  δq+q  ,K . N  2

l

q,q

(6.109)

K

In the commensurate case, this becomes     

 ˆ 1 )2 + m(Q ˆ 2 )2 + m(Q ˆ 3 )2 , M 2 = 4 m(Q

(6.110)

ˆ M ˆ for the 2Q, ˆ and M ˆ for 3Q ˆ as and we have the amplitude M1Qˆ for the 1Q, 2Q 3Q follows. 2A˜ Q , B1Q + B˜ 2Q

(6.111)

8A˜ Q , 2(B1Q + B˜ 2Q ) + B˜ 1QQ

(6.112)

6A˜ Q . B1Q + B˜ 2Q + B˜ 1QQ

(6.113)

M2 ˆ = − 1Q

M2 ˆ = − 2Q

M2 ˆ = − 3Q

From these expressions, we find that M1Qˆ < M2Qˆ < M3Qˆ ,

(6.114)

ˆ state is stable. when the 3Q It is common for the three structures that the condition A˜ Q < 0 is required. By comparing stability conditions (6.88), (6.101), and (6.106)–(6.107), and equilibrium free energies (6.90), (6.102), and (6.108), we obtain the magnetic phase diagram as shown in Fig. 6.11. In the AF phase (0 < B1Q + B˜ 2Q < |B˜ 1QQ |/2), the AF state is

176

6 Antiferromagnetism and Spin Density Waves

Fig. 6.11 Magnetic phase diagram for commensurate ˆ = 2π/a structures with |Q| for A˜ Q < 0 [111]. The first-kind antiferromagnetic ˆ and the 3Q ˆ (AF), the 2Q, phases are shown in the space of expansion coefficients B˜ 1QQ /B1Q and B˜ 2Q /B1Q , where B1Q > 0 for B˜ 2Q /B1Q > −1 and B1Q < 0 for B˜ 2Q /B1Q < −1

ˆ phase (0 < −B˜ 1QQ /2 < B1Q + B˜ 2Q < −B˜ 1QQ ), the only stable structure. In the 2Q ˆ structures are stable, and the AF and 2Q F1Qˆ > F2Qˆ .

(6.115)

ˆ phase (0 < B˜ 1QQ /2 < B1Q + B˜ 2Q , 0 < −B˜ 1QQ < B1Q + B˜ 2Q ), all three In the 3Q commensurate structures are stable, and we find that F1Qˆ > F2Qˆ > F3Qˆ .

(6.116)

We observe that the MSDW states with higher multiplicity are always more stable than the other states with lower multiplicity. The physical reason for this fact is as follows. First consider the free energy for the MSDW states (6.86) without the ˆ state mode–mode coupling term (the term with B1QQ ). The free energy for the 3Q ˆ is three times smaller than that for the 1Q state, as seen from (6.90) and (6.108). This free energy gain is caused by the increase in amplitudes of local magnetic moments as seen from (6.111) and (6.113), and is characteristic of the itinerant electron system. In the localized model such a mechanism of energy gain is forbidden because of the constraint of the constant amplitudes of local magnetic moments, so ˆ state is realized. that only the 1Q When the mode–mode coupling term B˜ 1QQ is positive, it suppresses the increase ˆ state (see (6.111) and (6.113)). As a in amplitudes of local moments of the 3Q ˆ result, the 3Q MSDW is stable only when B˜ 1QQ is smaller than a critical value. In the band theory [114], three possible ground states are found for γ -Fe using the LMTO method and the von Barth–Hedin LDA potential: the first-kind AF, and ˆ and 3Q ˆ structures. It is found numerically that the 3Q ˆ structure is the most the 2Q

6.4 Phenomenological Theory of Magnetic Structure

177

ˆ and amplitudes M of magnetic moments of fcc Fe for Table 6.1 Energy differences E − E(3Q) ˆ 2Q, ˆ and 3Q ˆ MSDW states which are calculated by the LDA-DFT theory [114]. E (E(3Q)) ˆ 1Q, ˆ structure) denotes the ground state energy for each structure (3Q ˆ 1Q

ˆ 2Q

ˆ 3Q

ˆ (mRy) E − E(3Q)

3.3

0.8

0.0

M (μB )

1.18

1.24

1.24

stable among the three at the equilibrium lattice constant a = 6.8 a.u. As shown in Table 6.1, their results of the energy at T = 0 follow the inequality (6.116) obtained by the phenomenological theory. It is interesting to note that the amplitudes ˆ and 2Q ˆ structures were found to be the of their magnetic moments for the 3Q same and larger than that for the AF structure: M3Qˆ = M2Qˆ > M1Qˆ . According to (6.112) and (6.113), this implies that B1Q + B˜ 2Q = B˜ 1QQ /2; γ -Fe calculated by the ˆ boundary in the 3Q ˆ phase of band theory is located in the vicinity of the AF-3Q Fig. 6.11.

6.4.2 Multiple SDW with Incommensurate Wave Vectors We consider next the multiple spin density waves (MSDW) with three incommensurate wave vectors Q1 , Q2 , and Q3 . These wave vectors satisfy the condition q + q  + q  + q  = K for q, q  , q  , and q  being one of {Qn } (see (6.79)). The linearly polarized MSDW is one of the possible MSDWs. It is described by ml =

3    m(Qn ) eiQn ·R l + m∗ (Qn ) e−iQn ·R l ,

(6.117)

n=1

with

m(Qn ) = mx (Qn ), my (Qn ), mz (Qn ) eiαn

(n = 1, 2, 3).

(6.118)

Here mx (Qn ), my (Qn ), and mz (Qn ) (n = 1, 2, 3) are assumed to be real. α1 , α2 , and α3 are phase factors. Additionally we consider the case in which m(Q1 ), m(Q2 ), and m(Q3 ) are orthogonal to each other. The free energy for the linear MSDW has the same form as the commensurate case, i.e., (6.86). FL =

3  2  4    AQ m(Qi ) + (B1Q + B2Q )m(Qi ) i=1

+

(2,3)(3,1)(1,2)  (i,j )

2  2  B1QQ m(Qi ) m(Qj ) .

(6.119)

178

6 Antiferromagnetism and Spin Density Waves

We therefore obtain the same results in which A˜ Q , B˜ 2Q , and B˜ 1QQ have been replaced by AQ , B2Q , and B1QQ , respectively. The phase diagram is the same as Fig. 6.11 in which B˜ 2Q and B˜ 1QQ have been replaced by B2Q and B1QQ , and AF, ˆ and 3Q ˆ have been replaced by 1Q, 2Q, and 3Q, respectively. Thus we find 2Q, the 3Q state in a wide range in the magnetic phase diagram. It is found in the band calculations that there is the incommensurate 3Q MSDW solution with Q = (0.6, 0, 0)(2π/a), (0, 0.6, 0)(2π/a), and (0, 0, 0.6)(2π/a) in the fcc Fe at lattice constant 6.8  a  7.0 [115]. Furthermore it is verified to be stable as compared with the 1Q SDW state irrespective of the lattice constant, and the amplitude of the magnetic moment for the 3Q state is larger than that for the 1Q state. These results are consistent with those of the phenomenological theory obtained in the last and present subsections; the 3Q MSDW is always stabilized when the 3Q solution exists, and has a larger amplitude of the magnetic moment as compared with the 1Q and 2Q SDW. Alternative MSDW are the helically polarized SDWs whose magnetic moments are described by (6.117) with m(Qj ) =

|m(Qj )| (ej k − iej m ) √ 2



(j, k, m) = (1, 2, 3)(2, 3, 1)(3, 1, 2) . (6.120)

Here ej k and ej m are unit vectors being perpendicular to the wave vector Qj , and orthogonal to each other (see (1.94) and (1.96)). The free energy for the helical MSDW is given as follows. FH =

3  2 4     AQ m(Qi ) + B1Q m(Qi ) i=1

+

(2,3)(3,1)(1,2) 

 2  2 (B1QQ + B2QQH )m(Qi ) m(Qj ) .

(6.121)

(i,j )

Note that the coefficient B2Q of the linear MSDW in (6.119) disappears and an additional coefficient B2QQH appears in the mode–mode coupling term in the case of the helical MSDW. However, the free energy (6.121) is again identical to (6.86) in which A˜ Q , B1Q + B˜ 2Q , and B˜ 1QQ have been replaced by AQ , B1Q , and B1QQ + B2QQH , so that we can easily obtain the results for the helical MSDW by exchanging the coefficients in the results of the last subsection. By comparing the free energy among the incommensurate linear 1Q, 2Q, 3Q MSDW, and the helical 1Q, 2Q, 3Q MSDW, we obtain the phase diagram for the incommensurate MSDW. An example is shown in Fig. 6.12. Note that the condition that both the linear and helical SDWs are stable is AQ < 0 and B1Q > 0. We find that the 3Q linear (3Q) and 3Q helical (3QH) MSDWs occupy most of the region −2 < B1QQ /B1Q < 1. This arises from the fact that the 3Q states are stable when the mode–mode coupling term B1QQ or B1QQ + B2QQH is relatively small. Since the stability of the helical 3Q is accompanied by the increase in amplitude

6.4 Phenomenological Theory of Magnetic Structure

179

Fig. 6.12 Magnetic phase diagram for the incommensurate SDWs for the expansion coefficients AQ < 0, B1Q > 0 and B2QQH /B1Q = 1 [112]. The phases of the 1Q linear SDW (1Q), the 2Q linear MSDW (2Q), the 3Q linear MSDW (3Q), the 1Q helical SDW (1QH), the 2Q helical MSDW (2QH), and the 3Q helical MSDW (3QH) are shown in the space of expansion coefficients B1QQ /B1Q and B2Q /B1Q . Coexistence lines between the linear and helical SDWs are indicated by solid lines. The phases shown with parenthesis are metastable states in which the ground state cannot be determined within the fourth-order Ginzburg–Landau theory. The gray regions indicate that the magnetic moment amplitude in the phase becomes so large that the fourth-order theory is not applicable

of the magnetic moment as shown in (6.114), it is characteristic of the itinerant electron magnetism. In the Heisenberg model, only the 1Q helical structure is stabilized by the competition between intersite exchange interactions. Experimentally, the 3Q MSDW is considered to be realized in γ -Fex Mn1−x (0.4 < x < 0.8) alloys [103]. Neutron diffraction experiments on the cubic γ -Fe100−x Cox (x < 4) alloy precipitates in Cu show a magnetic satellite peak at wave vector Q = (0.1, 0, 1)2π/a [102]. The magnetic structure was suggested to be a helical SDW, but has not yet been determined precisely. This is because the neutron diffraction analysis cannot distinguish between 1Q and 3Q states when the crystal structure of the γ Fe precipitates is properly cubic and the distribution of domains is isotropic. The present result based on the Ginzburg–Landau phenomenological theory tells us that the 3Q helical MSDW is always stable as compared with the 1Q and 2Q SDWs when the 3Q solution exists. It is desirable to investigate the relative stability be-

180

6 Antiferromagnetism and Spin Density Waves

tween 3Q linear and helical MSDWs in the first-principles ground-state calculations of γ -Fe. The same type of the helical 3Q MSDW has been found to be possible in MnSi system [116]. There we must take into account the spin–orbit coupling called the Dzyaloshinskii–Moriya interaction which appears in the noncentrosymmetric crystal structure.

Chapter 7

Magnetism in Dilute Alloys

Dilute magnetic alloys such as the transition metals dissolved in a simple metal show a unique magnetic property. In these alloys the oscillating long-range interactions appear via the conduction electrons, and the spin glass state in which the local magnetic moments are randomly oriented forms due to the competition between longrange ferro- and antiferro-magnetic interactions. We present the theoretical aspects of these properties in Sect. 7.1. In Sect. 7.2, we deal with the impurity limit. There the impurity magnetic moment disappears at low temperatures due to the formation of the singlet state. Because of the crossover from the local moment behavior to the Fermi liquid behavior with decreasing temperature, various anomalies called the Kondo effects appear. We briefly describe the formation of the singlet state known as the Kondo singlet.

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys The noble metal based alloys containing less than several percent of magnetic transition metals such as Cu–Mn and Au–Fe alloys form the substitutional disordered alloys. Susceptibilities of these alloys show a cusp at a temperature Tg indicating a magnetic phase transition as shown in Fig. 7.1. A remarkable point is that Tg showing a phase transition continues up to 0.005 at% impurity concentration in many cases [117]. The average inter-site distance between magnetic impurities at 0.005 at% is about 30 times that of the nearest-neighbor. Such a long-range magnetic interaction is explained neither by the super-exchange interaction nor by the direct exchange interaction as discussed in Sect. 1.6 because both interactions are caused by the overlap between neighboring atomic orbitals. In this section, we derive such a long-range magnetic interaction and briefly discuss the magnetism of dilute alloys. Let us consider the Cu–Mn dilute alloys as an example to derive the model Hamiltonian. The electronic structures of constituent Cu and Mn atoms are given by 3d10 4s and 3d5 4s2 , respectively. When these atoms form dilute alloys, electron Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_7, © Springer-Verlag Berlin Heidelberg 2012

181

182

7

Magnetism in Dilute Alloys

Fig. 7.1 Susceptibilities of Au–Fe dilute alloys as a function of temperature [118]

hoppings to the 3d orbitals of Cu atoms are suppressed because these orbitals form almost closed shells even in alloys. Electron hoppings between 4s orbitals must be fast since these atomic wave functions are extended in space. Since 3d orbitals on Mn atoms form unfilled shells, electron hoppings from the 3d orbitals to the 4s orbitals on the surrounding Cu atoms and the hopping backs to the Mn 3d orbitals are possible. We can neglect the direct electron hoppings between 3d orbitals on different Mn atoms because we are considering the dilute alloys less than 10 at% Mn. The Hamiltonian of the system is therefore described by the following two-band model. H = Hs + Hsd + Hd ,   † εs niσ + tij ciσ cj σ , Hs = iσ

Hsd =

ij σ

 iσ

(7.2)

ij σ

 (sd) †

(ds) † tij ciσ aj σ + tij aiσ cj σ ,

Hd =

(7.1)

εd ndiσ +



U ndi↑ ndi↓ .

(7.3) (7.4)

i (sd)

Here εs (εd ) is the atomic level of the s (d) orbital. tij (tij ) is the transfer integral between the s orbital on site i and s (d) orbital on site j . We assumed a single d orbital instead of five d orbitals on each magnetic atom for simplicity. Moreover we have introduced the on-site Coulomb interaction U between d electrons because the d electrons are localized as compared with the 4s electrons.

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys

183

Since 4s electrons form a wide band, the momentum representation may be the atomic more suitable. We can constructthe Bloch wave  function ϕk from √ 4s wave functions φi as ϕk = j φj j |k = j φj exp(−ik · R j )/ L. Making use of the Bloch functions  {ϕk }, we can diagonalize the Hamiltonian matrix (H s )ij = εs δij + tij (1 − δij ) as ij k|i(H )ij j |k   = εk δkk  . Accordingly we can † define the creation (annihilation) operator ckσ (ckσ ) in the momentum representa †  † tion, as ckσ = i ciσ i|k (ckσ = i ciσ k|i). The 4s conduction band Hamilto † nian Hs is then expressed as Hs = kσ εk nkσ . Here nkσ = ckσ ckσ is the number operator for 4s electrons with momentum k and spin σ . The hybridization term Hsd is also written by the creation and annihilation operators of conduction electrons in the momentum representation, so that the total Hamiltonian (7.1) is expressed as follows.  

† εk nkσ + aj σ + e−ik·Rj Vdk aj†σ ckσ eik·Rj Vkd ckσ H= kσ

+

 iσ

j kσ

εd ndiσ +



(7.5)

U ndi↑ ndi↓ .

i

∗ ) is the hybridization matrix element defined by Here Vkd (= Vdk  Vkd = ϕk∗ (r)V (r)φd (r) dr,

(7.6)

and V (r) is the atomic potential for an electron on the impurity site. The Hamiltonian (7.5) for dilute alloys is referred as the Anderson lattice Hamiltonian. Note that the Hamiltonian is derived without using the translational symmetry of magnetic impurities, so that it is applicable to disordered alloys. The Hamiltonian for the system with only one impurity at the origin is a special case of (7.5) and is given by  

 † † Vdk adσ εk nkσ + ckσ + Vkd ckσ adσ + εd ndσ + U nd↑ nd↓ . (7.7) H= kσ

kσ

σ

Here we omitted the site index 0 in the subscripts for simplicity and expressed the † creation (annihilation) operator for d electron as adσ (adσ ) to make the d character clearer. The Hamiltonian is called the Anderson model for the magnetic impurity in conduction band [119]. The Anderson lattice Hamiltonian has the eigen states {|{nkσ }, {ndj = 1, sj z }} in the zero-mixing limit (i.e., Vkd = 0) and at half-filling. Here {nkσ } denotes a configuration of the Fermi sea states for conduction electrons, while {ndj = 1, sj z } denotes the atomic states of magnetic impurities with the d electron number ndj = 1 and ‘local magnetic moment’ sj z (= ±1/2). Even for the system with a finite hybridization, the local magnetic moments on the impurity atoms should remain and may couple with the conduction electrons via the hybridization Vdk and Vkd . In order to obtain such a picture, we derive an effective Hamiltonian assuming a small hybridization.

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Fig. 7.2 Two possible virtual excitations via hybridization parameters Vkd and Vdk

We apply the second-order perturbation formula based the projection technique, i.e., (1.80) to obtain the effective Hamiltonian:

−1 0 − EP0 QHI P . Hp = P H P − P HI Q EQ

(7.8)

Here we adopt the Hamiltonian in the zero-mixing limit as the noninteracting Hamiltonian H0 , and treat the hybridization term as an interaction HI assuming small parameters: HI =



† eik·Rj Vkd ckσ aj σ + e−ik·Rj Vdk aj†σ ckσ .

(7.9)

j kσ

  εk nkσ + iσ εd ndiσ + The eigen values of H0 are given by Eλ0 = kσ  i U ndi↑ ndi↓ . Corresponding eigen states are given by {|{nkσ }, {ndj , sj z }}. We choose the {ndj = 1} states as subspace P on which the effective Hamiltonian Hp operates, and define the projection operator P such that P =  |{n }, {n = 1, sj z }{nkσ }, {ndj = 1, sj z }|. Then we have P H P = {n kσ }{sj z } kσ  dj P ( kσ εk nkσ + j εd )P . When we apply HI in subspace P, we have two kinds of excited states as shown in Fig. 7.2, the states with an empty d site and a particle above the Fermi level in the conduction band which is caused via Vkd , and the states with a doubly-occupied d site and a hole below the Fermi level in the conduction 0 − E 0 are ε − ε band, which is caused via Vdk . Associated excitation energies EQ k d P for the former and εd + U − εk for the latter, respectively. After the virtual excitations, the system returns to the original state via operator P HI . Thus the second term of (7.8) is written as

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys

185

−1 0 P HI Q EQ − EP0 QHI P   ik·Rj −ik  ·Rj e Vdk  Vkd † † =P alσ  ck  σ  ckσ aj σ ε − ε k d   j lkk σ σ

  eik  ·Rl −ik·Rj Vk  d Vdk † † + ck  σ  alσ  aj σ ckσ P . εd + U − εk  

(7.10)

j lkk σ σ

Arranging the expressions, we finally obtain the effective Hamiltonian as follows.    Vdk Vkd    HKL = εd + + εk nkσ + ei(k −k)·Rj Wk  k ck† σ ckσ εd − εk  kσ

−



j

e

k

i(k  −k)·R

j

Jk  k

j kk 



j kk

S j · ck† α (σ )αγ ckγ .

(7.11)

αγ

Here we have omitted the projector P for simplicity assuming that the Hamiltonian operates on the subspace P. The potential scattering parameter Wk  k and effective exchange energy parameter Jk  k between the local magnetic moment S j and conduction electron spins σ are defined by   1 1 1 , (7.12) − Wk  k = Vk  d Vdk 2 εk − εd − U εd − εk   Jk  k = Vk  d Vdk

 1 1 . + εk − εd − U εd − εk 

(7.13)

The effective Hamiltonian HKL consisting of the conduction band and the local magnetic moments is called the Kondo lattice Hamiltonian, and the exchange coupling between the local magnetic moment S j and conduction electrons is known as the Kondo exchange coupling. We can omit the potential scattering term and the constant term of d levels for magnetic phenomena. Moreover we neglect the momentum dependence of the Kondo coupling by assuming the constant hybridization parameter Vdk ≈ V and by taking the average of the coupling Jk  k with respect to the conduction band energy. We obtain then a simplified Kondo lattice Hamiltonian as follows. HKL =



εk nkσ −

kσ

JK  i(k  −k)·Rj  e S j · ck† α (σ )αγ ckγ . L  αγ

(7.14)

j kk

Here S j is the local magnetic moment on the magnetic impurity site j , which is  given by S j = αγ aj†α (σ )αγ aj γ /2. The Kondo coupling constant JK (= Jk  k ) is expressed as follows.   f (εk − μ) 1 − f (εk − μ) 2 . (7.15) + JK = |V | εk − εd − U εd − εk k

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Here f (εk − μ) is the Fermi distribution function, and μ is the chemical potential. Note that JK < 0 for εd  εk ∼ μ  εd + U (case of the strong U and the half filling). Using the local-orbital representation of the conduction electrons, we obtain the Kondo lattice Hamiltonian in the real space as follows.   εk nkσ − 2JK Sj · sj . (7.16) HKL = kσ

j

Here s j is the spin density of conduction electrons on site j , given by s j =  † αγ cj α (σ )αγ cj γ /2. The Kondo lattice Hamiltonian manifests the interaction between the conduction electrons and magnetic impurities. It is useful for analyzing the transport properties and the spin polarization phenomena of dilute magnetic alloys. In the dilute limit, the Hamiltonians (7.14) and (7.16) reduce to the following Hamiltonian, which is called the Kondo Hamiltonian or the sd Hamiltonian. HK =

 kσ

εk nkσ −

JK   S · ck† α (σ )αγ ckγ . L  αγ

(7.17)

kk

The sd Hamiltonian was often used to clarify thermal and transport properties of magnetic alloys in the dilute limit [120]. In the Kondo lattice Hamiltonian, the degree of freedom for conduction electrons remains so that the nature of magnetic interactions between magnetic impurities is not clear in this form. In the following, we consider the sub-subspace in which conduction electrons are in the ground state and only the degree of freedom on the local magnetic moments {S j } remains. To avoid confusion, we redefine the subspace P0 for the Kondo  lattice Hamiltonian by {|{nkσ }, {ndj = 1, sj z }} and the projector P0 by P0 = {nkσ }{sj z } |{nkσ }, {ndj = 1, sj z }{nkσ }, {ndj = 1, sj z }|. Then we define  the sub-subspace P by {|{nkσ }G , {ndj = 1, sj z }} and the projector P by P = {sj z } |{nkσ }G , {ndj = 1, sj z }{nkσ }G , {ndj = 1, sj z }|. Here {nkσ }G denotes the ground-state Fermi-sea configuration. We again make use of the projection technique for the second order perturbation, i.e., (7.8) to obtain the effective Hamiltonian in sub-subspace P where only the degree of freedom for magnetic impurity spins remains. We adopt the Kondo coupling term in (7.14) as the interaction part HI , and express it as follows.    † JK  i(k  −k)·Rj Sj − ck† ↑ ck↓ + Sj + ck† ↓ ck↑ + Sj z e σ ck  σ ckσ . (7.18) HI = − L  σ j kk

Here Sj ± are  the spin-flip operators defined by Sj ± = Sj x ± iSjy . We have then P HKL P = occ kσ εk . When we apply HI to P in sub-subspace P, we have three kinds of excited states: spin-flipped state sj z + 1 accompanied by the particle–hole pair excitations in conduction band via the Kondo coupling JK as shown in Fig. 7.3,

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys

187

Fig. 7.3 Three possible virtual excitations via the Kondo coupling JK

a spin-flipped state sj z − 1 accompanied by the particle–hole pair excitations in conduction band, and the non spin-flipped state sj z accompanied by the particle–hole pair excitations in conduction band. Associated excitation energies are now εk  − εk . Taking into account the possible paths via P HI , we obtain the effective Hamiltonian in which the degree of freedom of conduction electrons has been eliminated as follows.  occ    1 J (0)S(S + 1) − εk − J (R i − R j )S i · S j P . (7.19) HRKKY = P 2 2 k

j

(i,j )

Here J (R i − R j ) is defined by  J (R i − R j ) = 2

JK L

2  kk 

  nk − nk  cos k − k  · (R i − R j ) , εk  − εk

(7.20)

and nk denotes the electron occupation number per spin of conduction electrons with momentum k (i.e., the Fermi distribution function at T = 0 for electrons with momentum k). The effective Hamiltonian (7.19) has a form of the Heisenberg model as follows when we omit the constant terms.  J (R i − R j )S i · S j . (7.21) HRKKY = − (i,j )

Here the projector P has been omitted by promising that the Hamiltonian is applied on the subspace of the magnetic moments. Equation (7.21) is called the RKKY (Ruderman–Kittel–Kasuya–Yosida) interaction, and the coupling constant J (R) is known as the RKKY interaction coupling constant [121–123].

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7

Magnetism in Dilute Alloys

The RKKY interaction between the magnetic moments of impurity atoms is caused by a polarization of conduction electrons due to the local magnetic moments via the Kondo coupling. In fact, the RKKY interaction is expressed by the susceptibility of conduction electrons as follows. J (R i − R j ) =

JK2  χ(q) cos q · (R i − R j ). L q

(7.22)

Here χ(q) is the generalized static susceptibility defined by (6.33), i.e., m(q) = χ(q)h(q), and has been given as follows (see (6.36)). χ(q) =

2  nk+q − nk . L εk − εk+q

(7.23)

k

The RKKY interaction is obtained by a simple interpretation of the Kondo lattice Hamiltonian. According to the real-space representation of the Kondo lattice Hamiltonian (7.16), the conduction electrons feel a ‘magnetic field’ hj = JK S j on site j . The field causes a conduction electron spin polarization mi = χij hj on site i according to the linear response. The local moment S i has the Kondo interaction −JK S i · mi = −JK2 χij S i · S j according to the Hamiltonian (7.16). Summing up the interactions with respect to all the pairs, we obtain the Hamiltonian (7.21) with a coupling constant J (R i − R j ) = JK2 χij . (7.24)  Substituting the Fourier transform χij = q χ(q) exp iq · (R i − R j ) into the above expression, we find that the coupling constant (7.24) is identical with (7.22). The RKKY interaction is long-range since it is based on the interaction between the local magnetic moments via conduction electron polarization. In order to understand its long-range nature, let us consider the free electron model. In this case, the susceptibility (7.23) is expressed as    4Ω 1 1 χ(q) = dk + . (7.25) (2π)3 |k|
(7.27)

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys

189

Note that F (x) is an even function, F (0) = 2, and F (x) → 2/3x 2 (x → ∞). Using the integral (7.26), we obtain   q 1 . (7.28) χ(q) = χPauli F 2 2kF Here χPauli = 2ρ(εF ) is the Pauli uniform susceptibility in the free electron model, and ρ(εF ) is the density of states at the Fermi level εF . Substituting the expression (7.28) into (7.22) and extending the integration range from the first Brillouin zone to the whole region for convenience, we obtain J 2 Ω χPauli J (R) = K 2 (2π) R

 0

∞



 q dq qF sin qR. 2kF

(7.29)

Since F (x) is an even function, the above expression is written as follows. J (R) =

JK2 Ω χPauli I (R), (2π)2 2iR

(7.30)

and the integral I (R) is given by  q I (R) = eiqR . dq qF 2kF −∞ 

∞



(7.31)

As shown in Appendix H, the integral I (R) is obtained as follows. I (R) =

πi (−2kF R cos 2kF R + sin 2kF R). kF R 3

(7.32)

Substituting (7.32) into (7.30), we obtain the RKKY interaction in the free electron model. J (R) = 6πne JK2 χPauli

−2kF R cos 2kF R + sin 2kF R . (2kF R)4

(7.33)

Here ne is the electron number per unit cell given by ne = kF3 Ω/π , Ω being the volume of the unit cell. Thus, the RKKY interaction oscillates with the wave length π/kF as a function of R and decays as 1/R 3 as shown in Fig. 7.4. Note that the nearest-neighbor distance and the second nearest-neighbor distance correspond to 2kF R = 6.95 and 9.82, respectively, when we assume ne = 1 and the fcc lattice bearing in mind the Cu–Mn dilute alloys. The RKKY interaction is of long range because of the 1/R 3 dependence. Oscillatory long-range interactions are expected to cause a complex magnetism in the dilute alloys. We consider here the RKKY interactions Jij between the Isingtype spins with S = 1/2 for simplicity.  The molecular field Hamiltonian acting on a spin on site i is given by Hi = − j Jij Sj z Siz . Therefore the thermal average

190

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Magnetism in Dilute Alloys

Fig. 7.4 The RKKY interaction J /J0 as a function of 2kF R. Here J0 = 6πne JK2 χPauli

of the magnetic moment on site i (mi  = 2Siz ) is given in the molecular-field approximation as    β mi  = tanh Jij mj  . 4

(7.34)

j

For small magnetic moments near the critical temperature, we can expand the r.h.s. of the above self-consistent equation as follows. mi  =

β Jij mj  + · · · . 4

(7.35)

j

Taking the configurational average, we obtain mi  ≈

β Jij mj  + · · · . 4

(7.36)

j

Here the upper bar denotes the configurational average on magnetic impurities, and we have decoupled the correlation between the interaction constant and the surrounding magnetic moment for simplicity. From (7.36), we may obtain the Curie temperature TC . TC =

1 Jij . 4 j

(7.37)

7.1 Magnetic Interactions and Spin Glasses in Dilute Alloys

191

In the disordered system, we have another transition temperature. Taking first square of both sides in (7.35) and next taking the configurational average, we find  2  β mi 2 ≈ Jij Jil mj ml  + · · · . (7.38) 4 lj

When mi  = 0 and we neglect the correlation between local magnetic moments on different sites, we have mj ml  = mi 2 δj l . Then (7.38) reduces to mi 2 ≈

 2  β Jij2 mj 2 + · · · . 4

(7.39)

j

Therefore we have one more transition temperature Tg at which mi 2 vanishes. Tg =

1 2 Jij . 4

(7.40)

j

From the simple analysis mentioned above, we find at least two order parameters in dilute magnetic alloys: mi  and mi 2 . The system is in the ferromagnetic state (F) when mi  = 0 and mi 2 = 0. It is in the paramagnetic state (P) when mi  = 0 and mi 2 = 0. In addition to these cases we can also consider the case mi  = 0 and mi 2 = 0.

(7.41)

This case may correspond to a state such that there is no magnetization but the randomly oriented spins remain (see Fig. 1.16). It is called the spin-glass state (SG). The SG may be realized when magnetic interactions compete each other in the disordered alloys and there is no long-range order being found as the magnetic Laue spots. The transition temperature Tg is called the SG temperature. The cusps in the susceptibilities found in Au–Fe dilute alloys as shown in Fig. 7.1 are considered to show the spin glass transition [124].   The coupling constants j Jij and j Jij2 in TC and Tg are expressed by the parameters for atomic configuration. Note that the summations at the r.h.s. in (7.37) and (7.40) are taken with respect to the magnetic impurity sites. In order to express these summations as the sum over all the lattice sites, we introduce an occupation number αi of the impurity atom A on site i which takes 1 when site i is occupied by the atom A and takes 0 otherwise. In order to distinguish the two kinds of summations, we write them in (7.37) and (7.40) as j ∈A . Then  j ∈A

Jij =

 j

αi αj Jij =



αi αj Jij .

(7.42)

j

AA ≡ α α is the probability of finding the type of atom A on site j when Here pij i j AA , we can express site i is occupied by the same atom A. Using the probability pij

192

7

Magnetism in Dilute Alloys

Fig. 7.5 Magnetic ordering temperature vs. concentration curves for Au–Fe alloys obtained by various methods [118]



Jij in (7.37) and Thus we have j

 j

Jij2 in (7.40) as TC =

 j

AA J and pij ij

 j

AA J 2 , respectively. pij ij

1  AA pij Jij , 4

(7.43)

1  AA 2 pij Jij . 4

(7.44)

j

Tg =

j

AA = c , c being the When there is no short-range atomic order, we have pij A A impurity concentration. Thus,

TC = cA J˜0 ,

(7.45)

√ Tg = cA J˜. (7.46)   Here J˜0 = ( j Jij )/4 and J˜ = ( j Jij2 )1/2 /4. Figure 7.5 shows the ordering temperature vs. concentration curve in Au–Fe alloys. The linear concentration dependence of TC at cFe > 12 at% Fe and the concentration dependence of Tg at cFe < 12 at% Fe are explained by (7.45) and (7.46), respectively. The theory of spin glasses (SG) in the dilute alloys has been developed as a part of the disordered spin systems in the insulators because their Hamiltonian can be mapped into the RKKY spin Hamiltonian (7.21). Details of the advanced SG theory as a model of statistical mechanics can be found in the books by Parisi et al. [125] and by Fisher and Hertz [126].

7.2 Magnetic Impurity in Noble Metals

193

Fig. 7.6 Susceptibility of pure Au () and a Au 0.0054 at.% Fe dilute alloy (•) as a function of temperature [129]

7.2 Magnetic Impurity in Noble Metals In the dilute limit, the RKKY interaction is infinitesimally weak, and a magnetic impurity system is realized in the nonmagnetic metal. Such a small system shows a unique property which is quite different from the other systems. Figure 7.6 shows the temperature dependence of the susceptibility in Au–Mn dilute alloys as an example. The impurity contribution follows the Curie law at high temperatures indicating the existence of a local magnetic moment on Mn atom. With decreasing temperature, the susceptibility tends to saturate around a few Kelvin, indicating the disappearance of local magnetic moment of the impurity atom. The low temperature behavior and the related ground state cannot be understood by a simple picture of local magnetic moment. In this section we will briefly explain the basic properties of magnetic impurity dissolved in a simple metal. The magnetic impurity is described by the Anderson Hamiltonian as derived in the last section.  

 † † H= Vdk adσ εk nkσ + ckσ + Vkd ckσ adσ + εd ndσ + U nd↑ nd↓ . (7.47) kσ

σ

kσ

In order to obtain a physical insight to the magnetic impurity in the noble metals, we consider first a simplified model in which the number of sites in the conduction band has been reduced from infinity to one. This is called the ligand model or the zero-band-width model [127, 128]. The Hamiltonian is written as  

 H= εl nlσ + V aσ† cσ + cσ† aσ + εd ndσ + U nd↑ nd↓ . (7.48) σ

σ

σ

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Magnetism in Dilute Alloys

Here εl denotes a ligand level, nlσ is the number operator of the ligand electrons with spin σ . We have omitted the subscripts d and l of the creation and annihilation operators for simplicity. We assume that the ligand level εl is higher than the d level εd , and the Coulomb energy U is large enough as compared with the difference Δ0 ≡ εl − εd ; U  Δ0 . Moreover we consider the case that the hybridization |V | is small as compared with the difference, so that |V |  εl − εd  U . When we put two electrons in this system, the doubly occupied state on the impurity site is suppressed due to the strong Coulomb repulsion U , so that the magnetic impurity with one electron is realized and the d electron hybridizes with the ligand electron via V . In the case that V = 0, we have a ligand electron (nl = 1) and a d electron (nd = 1), so that the ground-state energy is given by E = εl + εd and associated 4 states are degenerate; |nl↑ nl↓ nd↑ nd↓  = |1010, |1001, |0110, |0101. Alternatively, degenerate 4 states are chosen to be the eigen states of the total spin S = sd + sl.  (s) 

Φ = √1 a † c† − a † c† |0, ↑ ↓ ↓ ↑ 2  (t)  † Φ = a c† |0, ↑ ↑ 1

(7.49)

 (t) 

Φ = √1 a † c† + a † c† |0, ↑ ↓ ↓ ↑ 0 2  (t)  † Φ = a c† |0. ↓ ↓ −1 (t)

Here |Φ (s)  (|Φα ) denotes the singlet (triplet) state with total spin S = 0 (S = 1). When V = 0, we have additional configuration with no d electron.  (0)  Φ = c↑† c↓† |0. (7.50) (t)

(t)

(t)

The Hamiltonian matrix for the 5 states (Φ (0) , Φ (s) , Φ−1 , Φ0 , Φ1 ) is given as follows. √ ⎞ ⎛ 2V 0 0 0 √2εl ⎜ 2V εl + εd 0 0 0 ⎟ ⎟ ⎜ ⎜ (7.51) H =⎜ 0 0 0 ⎟ 0 εl + εd ⎟ ⎝ 0 0 ⎠ 0 0 εl + εd 0 0 0 0 εl + εd 0 state |Φ (0)  is not coupled to the triplet states |Φ (t)  because the Note that the d hybridization σ aσ† cσ which is operated on the d0 state creates the singlet state. The singlet state is coupled to the d0 state because of the same reason. Diagonalizing the {Φ (0) , Φ (s) } sub-block, we find the eigenvalues ' ) 1( (7.52) E = εl + εd + Δ0 ± Δ20 + 8|V |2 , 2

7.2 Magnetic Impurity in Noble Metals

195

Fig. 7.7 The energy spectra of the ligand model for V = 0 (left) and for V = 0 (right)

where Δ0 ≡ εl − εd . For small |V |2 /Δ0 , we have E = 2εl + 2|V |2 /Δ0 and εl + εd − 2|V |2 /Δ0 . In summary, we have (i) the ligand electron state    (0)   √ V   |V |2  Ψ = 1 − 2 Φ (0) + 2 Φ (s) , (7.53) 0 Δ0 with the energy E0 = 2εl + 2|V |2 /Δ0 and the total spin S = 0, (ii) the singlet state   √ V  (0)   (s)  |V |2  (s)   Ψ Φ =− 2 + 1− 2 Φ , (7.54) Δ0 Δ0 with the energy Es = εl + εd − 2|V |2 /Δ0 and the spin S = 0, and (iii) the triplet (t) (t) (t) states |Φ−1 , |Φ0 , |Φ1  with energy Et = εl + εd and spin S = 1. The eigenvalues of the ligand model with V = 0 and V = 0 are summarized in Fig. 7.7. The ligand model suggests that the ground state of the Anderson model with strong U is a singlet at half filling. This means that the magnetic dilute alloys lose the magnetic moment at the ground state even in the strong U limit. The second point is that there is a characteristic temperature T ∗ = 2|V |2 /Δ0 above which the magnetic moment of the impurity state recovers and thus the susceptibility follows the Curie law, because the triplet states are located above the singlet ground state and the former is higher than the latter by 2|V |2 /Δ0 . Let us now return to the Anderson model, and obtain the singlet ground state in large U limit. According to the singlet ground state (7.54), we assume the following ground state wave function of electrons [130].   occ  †

† |Ψ  = A |Φ0  + Bk ad↑ ck↑ + ad↓ ck↓ |Φ0  . (7.55) k

Here A is a normalization factor, and {Bk } are the amplitudes to be determined variationally. The first term at the r.h.s. of (7.55), |Φ0  is the Fermi sea state with no d electron, which is defined by |Φ0  =

occ   † †  ck↑ ck↓ |0. k

(7.56)

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7

Magnetism in Dilute Alloys

† † Fig. 7.8 Three states |Φ0  (left), ad↑ ck↑ |Φ0  (middle), and ad↓ ck↓ |Φ0  (right) in the variational wave function

It corresponds to the ligand electron state |Φ (0)  with no d electron in the ligand model. On the other hand, each state in the second term of (7.55) has one d electron and forms a singlet state consisting of a d electron and a conduction electron with momentum k. occ 

 † †  1 † 1 † † † † †

ck  ↑ ck  ↓ |0. ck↑ + ad↓ ck↓ |Φ0  = √ ad↑ ck↓ − ad↓ ck↑ |Φk  ≡ √ ad↑ 2 2 k  =k

(7.57) These states are depicted in Fig. 7.8. The wavefunction (7.55) with A = 1 is written with use of {|Φk } as follows. |Ψ  = |Φ0  +

occ √  2 Bk |Φk .

(7.58)

k

Note that {|Φ0 , |Φk } are orthonormal. The amplitudes Bk are determined from the variational principle of the energy, E=

Ψ |H |Ψ  . Ψ |Ψ 

(7.59)

Here Ψ |Ψ  = 1 + 2

occ 

Bk∗ Bk ,

(7.60)

k

Ψ |H |Ψ  = E0 + 2

occ 

Vdk Bk∗ + Vkd Bk k

+2

occ 

(E0 + εd − εk )Bk∗ Bk ,

(7.61)

k

occ

and E0 = 2 k εk is the energy for the conduction band. Taking the variation of the energy, we have

δΨ |H |Ψ  Ψ |Ψ  − Ψ |H |Ψ δΨ |Ψ  = 0.

(7.62)

7.2 Magnetic Impurity in Noble Metals

197

Calculating the variations δΨ |Ψ  and δΨ |H |Ψ , and substituting them into (7.62), we find the self-consistent equation for Bk as follows. (E + εk )Bk = Vdk .

(7.63)

Here E = E − E0 − εd , and the energy E − E0 is obtained from (7.59), (7.60), and (7.61) as follows.  occ ∗ 2 occ k Vkd Bk + 2 k Bk [Vdk + (εd − εk )Bk ]  . (7.64) E − E0 = ∗ 1 + 2 occ k Bk Bk Substituting V dk of (7.63) into the second term of the numerator in (7.64) and mul∗ tiplying 1 + 2 occ k Bk Bk to both sides, we find the following equation. E = −εd + 2

occ 

Vkd Bk .

(7.65)

k

Substitution of Bk from (7.63) into the above equation yields the self-consistent equation for E as follows. E = −εd + 2

occ  |Vkd |2 . E + εk

(7.66)

k

Note that the above equation reduces to the solution (7.52) in the ligand model limit (εk → εl ). Assuming the flat band with the band edge −D at the bottom for conduction electrons, (7.66) is expressed as follows. E = −εd + 2ρ(0)|V |2 ln

|E| . D

(7.67)

Here ρ(ε) is the density of states per spin for conduction electrons which is defined by ρ(ε) = k δ(ε − εk ). Moreover we assumed that the band width D is large as compared with |E|. We have a solution of (7.67) at E < 0 as seen from Fig. 7.9 in which the curves of both sides of (7.67) are depicted. Since |E| → 0 when |V | → 0, we obtain an approximate solution for a small |V | as   |εd | |E| = D exp − . (7.68) 2ρ(0)|V |2 The characteristic temperature associated with the formation of the singlet ground state is called the Kondo temperature; TK ≡ |E|. In the present case, it is given by   |εd | . (7.69) TK = D exp − 2ρ(0)|V |2

198

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Magnetism in Dilute Alloys

Fig. 7.9 Graphical analysis of the self-consistent equation (7.67) for E

Note that the Kondo exchange coupling constant in the Kondo Hamiltonian is given by (7.13):  Jkk  = Vk  d Vdk

 1 1 . + εk − εd − U εd − εk 

(7.70)

For large U and the symmetric case (εd = −U/2), we have a simplified expression of the Kondo exchange coupling JK (= Jk  k  < 0) as follows. JK = −

2|V |2 . |εd |

(7.71)

Here we have omitted the momentum dependence of the hybridization parameter. The Kondo temperature (7.69) is expressed in terms of the Kondo exchange coupling as follows.  TK = D exp

 1 . ρ(0)JK

(7.72)

In the same way, we can calculate the d electron number as nd  = 1 −

πTK . Δ

(7.73)

Here Δ = 2πρ(0)|V |2 . The expression indicates that nd  approaches 1 as TK → 0. When the magnetic field h is applied on the impurity atom, the impurity d level is modified from εd to εdσ = εd − hσ , so that the variational parameter Bk is expected to be spin-dependent. The wave function is then modified as 

 occ 

† † |Ψ  = A |Φ0  + Bk↑ ad↑ ck↑ + Bk↓ ad↓ ck↓ |Φ0  . k

(7.74)

7.2 Magnetic Impurity in Noble Metals Table 7.1 The Kondo temperatures in the typical dilute alloys [133]

199

Alloys

Cu–Cr

Cu–Mn

Cu–Fe

Au–Fe

Al–Mn

TK (K)

2

0.01

30

0.8

500

Taking the same steps as before we reach the self-consistent equation for the ground state energy. E(h) = −εd + 2

occ  k

|Vkd |2 . E(h) + hσ + εk

(7.75)

The impurity susceptibility χimp is obtained from −(∂ 2 E(h)/∂h2 )h=0 by using the above self-consistent equation as follows. χimp =

1 . TK

(7.76)

The results shows that the susceptibility is finite even in the strong Coulomb interaction limit because of the formation of the singlet state. The magnitude is scaled by the inverse Kondo temperature. As expected from the arguments so far, all the related physical quantities are scaled by the Kondo temperature TK . The Kondo temperatures for typical dilute magnetic alloys which are determined experimentally are given in Table 7.1. The finite-temperature properties of the susceptibility in the Anderson model have been obtained accurately by using the numerical renormalization group technique [131, 132]. Figure 7.10 shows the temperature dependence of the susceptibilities of the Anderson model for strong (case A) and weak (case B) Coulomb interactions. In the strong Coulomb interaction regime, the Curie constant defined by C = T χimp vanishes at T = 0 because the ground state is the singlet. With increasing temperature, the local magnetic moment recovers above TK , and the Curie constant C approaches to a constant value which is close to the atomic value 1/4 (i.e., S(S + 1)/3 with S = 1/2). When the temperature is further elevated, the Curie constant starts to decrease around a temperature U and approaches to the free orbital value 1/8 where the empty and doubly occupied states as well as the spin-up and spin-down states are excited equally on the impurity site. In the case of B where the Coulomb interaction is close to the value of the Hartree–Fock magnetic instability point U/πΔ = 1, the ground state is again singlet, so that the Curie constant again vanishes at the ground state. However the Kondo temperature is very large (TK ∼ D) in this case and the local moment regime does not appear any more as seen in Fig. 7.10. It should be noted that the Hartree–Fock approximation does not lead to the singlet ground state. The approximation tells us that the local magnetic moment is stabilized when ρd (0)U > 1 [119]. Here ρd (ε) is the d density of states per spin on the impurity site. Figure 7.10 suggests that this condition is qualitatively applicable only when the temperature is in the regime TK  T  U .

200

7

Magnetism in Dilute Alloys

Fig. 7.10 The impurity susceptibility of the symmetric Anderson model for U/D = 0.001; U/πΔ = 12.66 (A) and U/πΔ = 1.013 (B) [131]

The stability of the Kondo singlet is often influenced by the other degrees of freedom such as the inter-site interactions. For example, the RKKY interaction |JRKKY | becomes stronger with increasing impurity concentration. When the formation energy of the spin glass per magnetic atom exceeds the Kondo temperature TK , the spin glass state is stabilized with the appearance of local magnetic moments. The many-body problem in the dilute magnetic alloys is called the Kondo problem, and associated anomalies are called the Kondo effects. Historically the Kondo effect was first found in the resistivity [120]. Resistivity usually decreases as the temperature decreases as is well known as Matthiessen’s law. In fact it consists of the impurity scattering term and the phonon term. The former is independent of temperature, while the latter monotonically decreases with decreasing temperature. In the dilute magnetic alloys such as Cu–Mn and Au–Fe alloys, it was found that the resistivity shows a minimum at low temperatures. Kondo explained first the anomaly on the basis of the sd model (7.17) [120]. He showed that the second-order Born approximation to the resistivity R due to magnetic impurities contains the logarithmic term.   4JK ρ(0) T ln . (7.77) R = RB 1 + L D Here RB is the resistivity in the first Born approximation. D is the band width of conduction electrons. Although the logarithmic term explains the resistance minimum, it leads to the divergence of the resistivity at zero temperature, which is unphysical. Higher order perturbation expansions at finite temperatures again led to the divergence of the resistivity at T ∼ TK [134]. The same behaviors were also found in the susceptibility and specific heat. Yosida investigated the ground-state properties of the magnetic impurity, and found that the singlet state consisting of the localized electron and a conduction electron is stabilized at the ground state [135]. The binding energy was found to be TK . Low temperature properties of the sd model were examined by Wilson on the

7.2 Magnetic Impurity in Noble Metals

201

basis of the renormalization group theory [136]. He found that the ratio of the impurity susceptibility χimp to the Sommerfeld coefficient due to impurity Cimp /T , Cimp being the impurity contribution to the specific heat, is a temperature-independent constant RW (= 2): T χimp = RW . Cimp

(7.78)

The same result was obtained by Nozières on the basis of a phenomenological Fermi liquid theory [137]. Yamada and Yosida derived the relation exactly by means of the perturbation expansion approach to the symmetric Anderson model [138, 139]. Later the exact solution to the sd model was obtained by Andrei with use of the Bethe ansatz method [140]. The exact solution to the Anderson model was also obtained by Wiegmann [141], and Kawakami and Okiji [142] with use of the same method. The readers who are interested in the details of the Kondo problems are recommended to refer to the books by Yosida [143], and by Fulde [127]. The Kondo anomalies as the bulk properties are also found in rare-earth compounds containing Ce and U compounds. Low temperature specific heats of these compounds, for example, show a huge value which is typically 100–1000 times those of the usual metals, indicating the existence of electrons with heavy effective mass. These materials are called the heavy fermion system. Details on the topic are found in the books by Hewson [128] and by Fulde [144, 145].

Chapter 8

Magnetism of Disordered Alloys

Substitutional disordered alloys show a variety of magnetic properties with changes in composition and temperature, which clarify the formation of magnetic moments in metals. In this chapter, we present the theories of itinerant magnetism in disordered alloys, and clarify the magnetic properties of 3d transition metal alloys. We first overview in Sect. 8.1 the magnetization vs. concentration curves in transition metal alloys which are known as the Slater–Pauling curves, and present a simple picture based on the rigid band model and Friedel’s virtual bound state. In Sect. 8.2, we introduce the coherent potential approximation (CPA) to treat the configurational disorder within the single-site approximation (SSA). We then present the Hartree– Fock (HF) CPA theory at the ground state and extend the finite-temperature theory presented in Sect. 3.3 to disordered alloys. Since the SSA does not consider the inter-site correlations of the magnetic moments due to configurational disorder, we present in Sect. 8.3 the ground-state theory of the local environment effects (LEE) which goes beyond the HF-CPA. Next we present the theory of the LEE at finite temperatures. In the last Sect. 8.4, we extend the molecular dynamics theory presented in Sect. 6.3 to the disordered alloys to describe more complex system, and argue the magnetism of Fe–Cr alloys showing the ferromagnetism, the antiferromagnetism, and the spin glass.

8.1 Slater–Pauling Curves The substitutional disordered alloys containing 3d transition metals have been much investigated to understand the formation of metallic magnetism and to clarify the effects of disorder on their magnetism. Experimentally the ground-state magnetization vs. concentration curves in 3d transition metal alloys are well-known as the Slater–Pauling curves [146]. Understanding of the systematic change of their magnetizations has been one of the central issues of magnetism in disordered alloys. We briefly explain in this section a global behavior of the magnetism. Figure 8.1 shows the Slater–Pauling curves in various transition metal alloys. The curves consist of the line with the slope −45◦ from Cu to Fe which includes Ni–Cu, Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_8, © Springer-Verlag Berlin Heidelberg 2012

203

204

8

Magnetism of Disordered Alloys

Fig. 8.1 Slater–Pauling curves in transition metal alloys as a function of atomic number [146]

Ni–Co, Co–Fe, and Fe–Ni alloys, the 45◦ line from Fe to Cr which includes Fe–V and Fe–Cr, and the other branches showing a rapid decrease of magnetization with adding the second elements. The ground-state magnetization M per atom is given by the up-spin valence electron number N↑ minus the down-spin valence electron number N↓ : M = N↑ − N↓ . Since the total valence electron number Z is given by Z = N↑ + N↓ , the magnetization is expressed as M = 2N↑ − Z.

(8.1)

The zeroth approximation is to assume that electrons feel a common potential so that they occupy a common band as in the case of the pure metal. This is referred as the rigid-band model. In the rigid band model, Ni and Co alloys are considered to be strong ferromagnets in which the up spin band is filled as shown in the left of Fig. 8.2. In this case, N↑ is constant and the magnetization increases with decreasing Z according to (8.1). This explains the −45◦ line in the Slater–Pauling curves. It is interpreted that the Fermi level touches the top of the up-spin band around Z = 8.25 and the atomic level for down spin electron ε↓ starts to decrease due to the Hartree– Fock type potential ε↓ = ε0 + U N↑ , so that N↓ is kept constant with decreasing Z (see the right of Fig. 8.2). Then the magnetization decreases with decreasing Z along the 45◦ line according to the expression M = Z − 2N↓ as found in the Slater– Pauling curves of Fe–Cr and Fe–V alloys (see Fig. 8.1). For a more detailed explanation, we can take into account explicitly the concenA + (1 − c)N B , tration dependence of N↑ and Z as N↑ = Nd↑ + Nsp↑ , Nd↑ = cNd↑ d↑ and Z = cZA + (1 − c)ZB . Here Nd↑ (Nsp↑ ) is the d (sp) electron number per atom, A (N B ) is the d electron number for atom A (B), Z (Z ) is the valence elecNd↑ A B d↑ tron number of atom A (B), and c denotes the concentration of atom A. Substituting

8.1 Slater–Pauling Curves

205

Fig. 8.2 Schematic densities of states for the strong ferromagnet (left) and the weak ferromagnet (right)

Fig. 8.3 Systematic change in atomic d levels of 3d transition metals (left) and the virtual bound states above the Fermi level εF (right)

these relations into (8.1) and using the magnetization MB of the matrix B, we obtain the following expression.

  A B M = MB − c ZA − ZB − 2 Nd↑ . (8.2) − Nd↑ The Ni–Cu, Ni–Co, and Ni1−c Fec (c < 0.75) alloys are strong ferromagnets so A = N B = 5. Equation (8.2) then reduces to M = M − c(Z − 10). Thus that Nd↑ Ni A d↑ we again obtain the curve with the −45◦ slope in agreement with the experimental data shown in Fig. 8.1. The same argument is applicable to the Co–Ni and Co–Fe alloys. The concentration dependence of the magnetization in Ni–Cr alloys is quite different from those following the −45◦ line. Note that the atomic d level εd decreases with increasing the atomic number because the attractive interaction between electron and nucleus becomes stronger (see the l.h.s. of Fig. 8.3). Thus the difference in atomic level εd between Cr and Ni is expected to be large as compared with the d band width, so that an impurity bound state called the virtual bound state [147] may appear above the Fermi level as shown in the r.h.s. of Fig. 8.3. We have then

206

8

Magnetism of Disordered Alloys

A = 0 and N B = 5. The magnetization (8.2) is then expressed as Nd↑ d↑

M = MB − c(ZA − ZB + 10).

(8.3)

Thus we have M − MB = −6c for Ni–Cr, and M − MB = −5c for Ni–V. These results explain the branches at Ni in the Slater–Pauling curves (Fig. 8.1). In the same way we obtain M − MB = −7c for Co–Cr, and M − MB = −8c for Co–Mn. The results explain the branches at Co in the Slater–Pauling curves [148].

8.2 Single-Site Theory of Disordered Alloys Although a simple interpretation of the Slater–Pauling curves is presented in the last section and it is easy to understand the global behavior of the curves on the basis of a simple interpretation, the role of the electronic structure of disordered alloys behind the phenomena is not clear. In order to understand the magnetic properties of the alloys one must know their electronic structure from a microscopic point of view. Electronic structure calculations in disordered alloys, however, is not easy. The difficulty is that there is no translational symmetry in the system so that we cannot apply the Bloch theory. We present in this section the coherent potential approximation (CPA) [51] to calculate the electronic structure of alloys in the singlesite approximation, and demonstrate how the Hartree–Fock CPA theory can explain the Slater–Pauling curves [149, 150]. Furthermore we extend the finite-temperature theory presented in Sect. 3.3 to the disordered alloys. Let us consider the A–B substitutional binary alloys showing the ferromagnetism, which is described by the Hubbard model in the Hartree–Fock approximation as follows (see (2.7)).   † H= εiσ niσ + tij aiσ aj σ . (8.4) iσ

ij σ

Here εiσ = εi0 +U ni /2−U mi σ/2 is the Hartree–Fock atomic potential on site i. tij is the transfer integral between sites i and j . Note that the atomic level εi0 as well as the charge and spin densities (ni  and mi ) are now site-dependent. The local charge and magnetic moment on site i are given by   ρiσ (ω), (8.5) ni  = dω f (ω − μ) σ

 mi  =

dω f (ω − μ)



σρiσ (ω).

(8.6)

σ

Here f (ω) is the Fermi distribution function, and μ denotes the chemical potential. ρiσ (ω) is the density of states on site i for an electron with spin σ in the Hartree– Fock approximation. It is given by the Green function Giiσ (z) with z = ω + iδ as

8.2 Single-Site Theory of Disordered Alloys

207

Fig. 8.4 Schematic representation of the coherent potential approximation [151]. The central site is occupied by an atom α (= A or B). The hatched sites are occupied by a coherent potential Σσ (z).  AV at the l.h.s. means a configurational average on the central site

follows. 1 Im Giiσ (z), π   Giiσ (z) = (z − H σ )−1 ii . ρiσ (ω) = −

(8.7) (8.8)

Here H σ is the Hartree–Fock one-electron Hamiltonian matrix defined by (H σ )ij = εiσ δij + tij (1 − δij ). In the binary alloys, the atomic levels εiσ take a value: εAσ (εBσ ) when site i is occupied by atom A (B). We neglect the disorder of transfer integrals called the offdiagonal disorder. Because the atomic levels εiσ are random variables, the Bloch theory is not applicable for the calculation of the Green function. We make use of the coherent potential approximation (CPA) here to obtain the Green function with use of the locator expansion [151]. Introducing the locator matrix L by means of (L)ij = Li δij = (z − εi )−1 δij , we express the Green function matrix as G = (L−1 − t)−1 = (1 − L t)−1 L where t denotes the transfer integral matrix tij . Expanding the Green function with respect to t, we obtain   Gii (z) = Li + Li tij Lj tj i Li + Li tij Lj tj k Lk tki Li + · · · . (8.9) j =i

j =i k =j,i

Here we have omitted the spin suffix for simplicity. The r.h.s. of (8.9) consists of the contribution from all the paths which start from site i and end at the same site i. They are expressed as follows by using the sum of all the paths Si which start from site i end at site i without returning to site i on the way.

−1 . Gii (z) = Li + Li Si Li + Li Si Li Si Li + · · · = L−1 i − Si

(8.10)

In the above expression, all the information outside the central atom i is in the self-energy Si . To obtain the Green function Gii (z) in a single-site approximation, we approximate the random potentials on the surrounding sites with an energydependent coherent potential Σ(z) (see the l.h.s. of Fig. 8.4). Note that the same idea was used in the metal-insulator transition [9, 10] and the single-site theory of

208

8

Magnetism of Disordered Alloys

spin fluctuations (see Sects. 3.3 and 3.4). We have then an impurity Green function for atom α on site i from (8.10) as follows.

−1 Gασ (z) = L−1 . (8.11) ασ − Sσ Here L−1 ασ = z − εασ is the inverse locator on the central site with a type of atom α. Sσ is the self energy in which all the atomic levels have been replaced by the coherent potential Σσ (z). Note that we have omitted the site indices for simplicity and have recovered the spin suffix. The self-energy Sσ is obtained from the coherent Green function Fσ (z) in which all the sites are occupied by the coherent potential (see the r.h.s. of Fig. 8.4).

−1 Fσ (z) = Lσ−1 − Sσ . (8.12) Here Lσ−1 (z) = z − Σσ (z). Substituting Sσ obtained from (8.12) into (8.11), we obtain the impurity Green function as follows.

−1 −1 −1 Gασ (z) = L−1 . (8.13) ασ (z) − Lσ (z) + Fσ (z) Since the coherent Green function Fσ (z) is defined by Fσ (z) = [(z − Σσ (z) − t)−1 ]ii , it is obtained from the following formula.  ρ(ε) dε . (8.14) Fσ (z) = Lσ−1 (z) − ε Here ρ(ε) is the density of states (DOS) for the energy eigen values of transfer matrix tij . The coherent potential Σσ (z) (or coherent locator Lσ (z)) is obtained from the condition that the configurational average of the impurity Green function (8.13) should be identical with the coherent Green function (see Fig. 8.4).  cα Gασ (z) = Fσ (z). (8.15) α

Here cα denotes the concentration of atom α. The above equation is known as the CPA equation. The same type of self-consistent equation was obtained in (3.85). There the random potential was produced by thermal spin fluctuations, and the thermal average was taken at the l.h.s. of the CPA equation. When the off-diagonal disorder in transfer integrals is significant, we can take into account the effects assuming that the transfer integral tAB between atoms A and B is given by the geometrical mean of tAA and tBB . This means that the transfer integrals are expressed with use of the parameters ri and rj as tij = ri∗ tBB rj .

(8.16) √ Here |ri | = |tAA |/|tBB | for i = A and 1 for i = B. In this case the off-diagonal disorder reduces to the diagonal disorder by considering ri∗ Gii (z)ri ; ri∗ Gii (z)ri reduces to (8.9) in which {Li } have been replaced by {Li = |ri |2 /(z − εiσ )}. Equation

8.2 Single-Site Theory of Disordered Alloys

209

Fig. 8.5 Concentration dependences of average magnetic moment (AV: m), and the local moments of Fe and Cr atoms, mFe  and mCr  of Fex Cr1−x alloys [152]. The solid lines show the result calculated by the Hartree–Fock CPA theory with use of the model density of state depicted in the lower-left corner. Experimental data are shown by •, ◦ [153–155], ,  [156–158]

(8.13) is then replaced by

−1 −1 −1 Gασ (z) = |rα |−2 L−1 , ασ (z) − Lσ (z) − Fσ (z)

(8.17)

2 where L−1 ασ = (z − εασ )/|rα | . The CPA equation (8.15) should be replaced by  cα |rα |2 Gασ (z) = Fσ (z). (8.18) α

The atomic level εασ in the impurity Green function Gασ (z) is given by εασ = εα0 + U nα /2 − U mα σ/2. The impurity charge nα  and magnetic moment mα  are calculated from Gασ (z) via the Hartree–Fock equations (8.5) and (8.6) selfconsistently. In actual applications the Coulomb interaction U in the Hartree–Fock CPA theory has to be regarded as an effective interaction parameter which is renormalized due to electron correlations (see Sect. 2.2). In the self-consistent calculations, the noninteracting model DOS ρ(ε) in the coherent Green function (8.14) is usually taken from band calculations. The Hartree–Fock CPA theory explains the basic properties of the Slater–Pauling curves presented in Fig. 8.1. Figure 8.5 shows the magnetic moments vs. concentration curves of Fe1−x Crx alloys calculated by the theory [152]. Because of the single band model, we multiplied the results by a factor of 5, and compared them with the experimental data. The averaged magnetic moment monotonically decreases with increasing Cr concentration in agreement with the experimental data. The Fe magnetic moments are parallel to the bulk magnetization and build up the ferromagnetism. The atomic level of Cr is above the Fermi level, and the down-spin

210

8

Magnetism of Disordered Alloys

Fig. 8.6 Calculated densities of states of Ni1−c Crc [149, 150]

bands of Cr sites more hybridize with those of Fe sites. This increases the downspin states of Cr sites below the Fermi level. Thus, the average Cr magnetic moment is antiparallel to the bulk magnetization. The Cr atoms lose the magnetic moment around 70 at% Cr. These results explain the neutron data for the magnetic moments [154–157]. Figure 8.6 shows the calculated DOS in Ni1−x Crx alloys [149, 150]. In the small Cr concentrations, we find the virtual bound state above the Fermi level which was assumed in the last section. It develops with increasing Cr concentration. The strong ferromagnetism collapses around 5 at% Cr, and the alloy loses the average magnetic moment around 10 at% Cr in agreement with the experimental data as shown in Fig. 8.7. At finite temperatures we need to extend the single-site theory presented in Sects. 3.3 and 3.7 to the disordered alloys. Since the disordered alloys are complex system, we apply a simplified model; we adopt the 5-fold equivalent band model (i.e., 5 times the single band model). Furthermore we adopt the static approximation (i.e., the high-temperature approximation) and omit the transverse spin fluctuations. The free energy (6.52) is then written as

F = −β

−1

ln

  i



β J˜i dξi e−βE(ξ ) . 4π

(8.19)

In the above expression, J˜i is the effective exchange energy defined by J˜i = U0 /D + (1 − 1/D)J , D being the orbital degeneracy (D = 5). The effective potential E(ξ )

8.2 Single-Site Theory of Disordered Alloys

211

Fig. 8.7 Calculated magnetic moments of Ni and Cr as well as the average in Ni1−c Crc alloys [149, 150]. The experimental average moments are shown by  [159],  [160], and  [161]

is obtained from (6.53) as

1 

U˜ i n˜ i (ξ )2 − J˜i ξi2 . E(ξ ) = −β −1 ln tr e−βH (ξ ) − 4

(8.20)

i

Here the effective Coulomb interaction U˜ i is defined by U˜ i = U0 /D + (1 − 1/D)(2U1 − J ). n˜ i (ξ ) is the Hartree–Fock charge on site i when the ‘magnetic moments’ {ξj } are given (see (6.58)).   tr( m nim e−βH (ξ ) ) nim 0 = . n˜ i (ξ ) = tr(e−βH (ξ ) ) m

(8.21)

The Hamiltonian H (ξ ) is given as follows (see (6.57)).    1 1 † εi0 − μ + U˜ i n˜ i (ξ ) − J˜i ξi σ nimσ + H (ξ ) = tij aimσ aj mσ . 2 2 m iσ

ij σ

(8.22) In order to treat the disorder of alloys, we rewrite the energy potential E(ξ ) as    1 

D E(ξ ) = dω f (ω) Im tr ln(z − H ) − U˜ i n˜ i (ξ )2 − J˜i ξi2 . (8.23) π 4 i

Here the Hamiltonian matrix H is defined by   1 ˜ 1 ˜ 0 (H )iσj σ  = εi − μ + Ui n˜ i (ξ ) − Ji ξi σ δij + tij δσ σ  . 2 2

(8.24)

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Magnetism of Disordered Alloys

We assume again the geometrical mean for the transfer integral; tij = ri∗ tij0 rj . Introducing the r matrix by means of (r)iσj σ  = ri δij δσ σ  , we can express the integrand of the first term in (8.23) as



     Im tr ln(z − H ) = Im tr ln r † L−1 r − r † t 0 r = Im tr ln L−1 − t 0 . (8.25) Here the locator is defined by −1

z − εi0 + μ − 12 U˜ i n˜ i (ξ ) + 12 J˜i ξi σ  = L iσj σ  = L−1 δ δ δij δσ σ  . ij σ σ iσ |ri |2

(8.26)

Introducing an effective locator Lσ (z) into the r.h.s. of (8.25), we expand it with respect to the site as







 tr ln L−1 − t 0 = tr ln L −1 − t 0 + tr ln 1 + L−1 − L −1 F + tr ln 1 − t˜F  . (8.27) Here F and F  are the diagonal and the off-diagonal coherent Green functions defined by Fij σ = [(L −1 − t 0 )−1 ]iiσ δij and (F  )ij σ = [(L −1 − t 0 )−1 ]ij σ (1 − δij ), respectively. t˜ is the single-site t matrix defined by

−1 −1

 L − L −1 . (8.28) t˜ = − 1 + L−1 − L −1 F Substituting (8.27) into (8.23) and neglecting the nonlocal term (i.e., F  term), we obtain the free energy per site in the single-site approximation as follows. 

1 D Im tr ln L −1 (z) − t 0 FCPA = dω f (ω) N π    β J˜α dξ e−βEα (ξ ) , − β −1 cα ln (8.29) 4π α  Eα (ξ ) =

dω f (ω)

 

 D −1 Im ln 1 + L−1 ασ (z, ξ ) − Lσ (z) Fσ (z) π σ

1 1 − U˜ α n˜ α (ξ )2 + J˜α ξ 2 . 4 4

(8.30)

Here Fσ (z) is the coherent Green function given by (8.14). L−1 ασ (z, ξ ) is defined by the diagonal part of (8.26) in which site i is occupied by atom α. Lασ (z, ξ )−1 =

z − εα0 + μ − 12 U˜ α n˜ α (ξ ) + 12 J˜α ξ σ . |rα |2

(8.31)

The local charge on atom α, n˜ α (ξ ) at the r.h.s. of (8.30) and (8.31) is given by   n˜ α (ξ ) = dω f (ω) ρασ (ω, ξ ), (8.32) σ

8.2 Single-Site Theory of Disordered Alloys

213

and the density of states ρασ (ω, ξ ) is obtained from the Green function as ρασ (ω) = −

D Im Gασ (z, ξ ). π

(8.33)

The Green function Gασ (z, ξ ) is given by (8.17) in which Lασ (z)−1 has been replaced by (8.31). The effective medium Lσ−1 (z) is determined by the condition that the t matrix vanishes in average so that the nonlocal correlations at the r.h.s. of (8.27) become minimum.   t˜(z, ξ ) ≡ cα dξ pα (ξ ) t˜α (z, ξ ) = 0. (8.34) α

Here pα (ξ ) is the probability of finding a ‘magnetic moment’ ξ on atom α, and is given by pα (ξ ) = 

e−βEα (ξ ) . dξ e−βEα (ξ )

(8.35)

Note that (8.34) is equivalent to the following equation, which is known as the CPA equation. 

 cα

dξ pα (ξ )|rα |2 Gασ (z, ξ ) = Fσ (z).

(8.36)

α

This is the extension of the Hartree–Fock CPA equation (8.18) to the finite temperature [162], and it reduces to the Hartree–Fock CPA equation (8.18) when T → 0. Equations (8.32) and (8.36) form the self-consistent equations for n˜ α (ξ ) and Lσ−1 (z). The local charge and magnetic moment of atom α are obtained as (see (3.258) and (3.259))  nα  = n˜ α (ξ ) =

dξ pα (ξ )n˜ α (ξ ),

(8.37)

dξ pα (ξ ) ξ.

(8.38)

 mα  = ξα  =

In the actual calculations, it is convenient to introduce a charge neutrality potential {wi (ξ )}. The potential due to the spin polarization is defined by

1 wi (ξ ) = U˜ i n˜ i (ξ ) − n˜ i (0) . 2

(8.39)

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Magnetism of Disordered Alloys

With use of the relation n˜ i (ξ ) = 2wi (ξ )/U˜ i + n˜ i (0), we can rewrite the effective potential as

E(ξ ) = −β −1 ln tr e−βH (ξ )   1 1 1 ˜ 2 2 2 ˜ − Ui n˜ i (0) + wi (ξ ) n˜ i (0) + wi (ξ ) − Ji ξi , 4 4 U˜ i i       1 ˜ † H (ξ ) = ε˜ i + wi (ξ ) − Ji ξi σ nimσ + tij aimσ aj mσ . 2 m iσ

(8.40)

(8.41)

ij σ

Here the atomic level in the nonmagnetic state is defined by ε˜ i = εi0 − μ + U˜ i n˜ i (0)/2. Eliminating the constant term U˜ i n˜ i (0)2 /4 from (8.40), and taking the large Coulomb interaction limit (U˜ i → ∞), we obtain the effective potential as  

 1 wi (ξ ) ni − J˜i ξi2 . (8.42) E(ξ ) = −β −1 ln tr e−βH (ξ ) − 4 i

Here ni = n˜ i (0) is the d electron number of each atom which is usually taken to be the value of the constituent metal in the pure limit because of the charge neutrality on each atom. The self-consistent equation to determine wi (ξ ) is obtained from (8.39) with U˜ i → ∞ as follows. ni = n˜ i

(3 4) 1 ε˜ i + wi (ξ ) − J˜i ξi σ . 2

(8.43)

The above equation implies that we can determine the charge potential εi + wi (ξ ) from the charge neutrality condition on each site. Equation (8.43) reduces the number of adjustable parameters such as the atomic levels and the effective Coulomb interaction parameters U˜ α , and allow us to determine automatically the chemical potential. When we adopt the charge neutrality potential, the energy potential (8.30) should be replaced by   

 D −1 Eα (ξ ) = dω f (ω) Im ln 1 + L−1 ασ (z, ξ ) − Lσ (z) Fσ (z) π σ 1 − wα (ξ ) nα + J˜α ξα2 . 4

(8.44)

The effective inverse locator L−1 α (z, ξ ) in (8.36) and (8.44) should be replaced by −1 Lασ (z, ξ ) = (z − ε˜ α − wα (ξ ) + J˜α ξ σ/2)/|rα |2 . The charge potentials wα (ξ ) are determined by the charge neutrality condition (8.43) in which the suffix for site i has been replaced by the atom α.

8.2 Single-Site Theory of Disordered Alloys

215

Fig. 8.8 Concentration dependence of magnetic moments at T = 0 K in Fec Ni1−c alloys [162]. The solid curves are the local moments for each atom. The dot-dashed curve is the total magnetization. The dashed curve shows the magnitude of the local moment on Fe site in the paramagnetic state. The arrows means the transition to the paramagnetic state. Experimental data are shown by ◦ (average magnetization),  (Fe magnetic moment), and  (Ni magnetic moment) [163, 164]

The Fec Ni1−c alloys form the fcc lattice in the range 0 ≤ c < 0.7, and show the ferromagnetic instability around 65 at% Fe where various anomalies called the Invar effects occur. We show in Fig. 8.8 the concentration dependence of magnetic moments calculated by the single-site theory of disordered alloys as an example [162]. With increasing Fe concentration, the Ni local moment (LM) hardly changes due to strong ferromagnetism, while the Fe LM gradually decreases. Calculated results explain the experimental data obtained by the neutron scattering up to 65 at% Fe. Around c = 0.6, the Fermi level reaches the top of the up-spin d band so that the instability of the ferromagnetism is induced. The single-site theory predicts the firstorder transition from the ferromagnetic state to the paramagnetic state at c∗ = 0.65, while the experimental data indicate the second-order transition. It is remarkable that the Fe LM remains even beyond c∗ as shown in the figure. It suggests the existence of an ordered state after collapse of the ferromagnetism. Experimentally, the spin glass state is found beyond c∗ [163], which will be discussed in the next section. Figure 8.9 shows the concentration dependence of the Curie temperature calculated by the single-site theory of disordered alloys. The Curie temperature first increases with increasing Fe concentration, and shows the maximum around 20 at% Fe. It implies a strong ferromagnetic coupling between Fe and Ni LM as compared with Ni–Ni and Fe–Fe couplings. The result explains the experimental curve of TC qualitatively. The temperature dependences of Fe and Ni LM are shown in Fig. 8.10. The Fe LM in pure Ni (i.e., c = 0) hardly changes with increasing temperature and rapidly decreases near TC , indicating a strong magnetic coupling between Fe and Ni LMs. On the other hand, both curves for Fe0.5 Ni0.5 alloys almost linearly decrease near TC . The result is consistent with the downward deviation from the S = 1/2 Brillouin curve near c∗ found in the experimental data.

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Magnetism of Disordered Alloys

Fig. 8.9 Concentration dependence of the Curie temperature in Fec Ni1−c alloys [162]. The solid (dashed) curve is the calculated result (observed one [146])

Fig. 8.10 Temperature dependences of the local magnetic moments of each atom [162]. The Fe concentration is denoted by c

Calculated paramagnetic spin susceptibility follows the Curie–Weiss law except in the case of pure Ni in which the inverse susceptibility shows upward convexity. Calculated effective Bohr magneton number monotonically increases with increasing Fe concentration as shown in Fig. 8.11, and the Weiss constant becomes negative beyond c = 0.65 in agreement with the experimental data.

8.3 Theory of Local Environment Effects in Magnetic Alloys

217

Fig. 8.11 Calculated effective Bohr magneton number meff (dashed curve) and the Weiss constant Θ (solid curve) [165]. Experimental values are shown by  [166] and  [167], respectively

8.3 Theory of Local Environment Effects in Magnetic Alloys The magnetic properties of disordered alloys are often influenced strongly by the local electronic and magnetic configurations. In particular magnetic moments in the system with holes in the up-spin electron band change their magnitudes and directions depending on their local environments. The behaviors associated with the change of local electronic and magnetic states due to surrounding atomic configurations are known as the local environment effects (LEE). Since the effects are related with the intersite correlations, they are not described by the single-site theories presented in the last section. One needs a theory beyond the CPA. In this section we will present the theories of the LEE at zero temperature [168, 169] as well as at finite temperatures [170–172], and elucidate the LEE in the magnetism of Ni–Cu, Ni–Mn, and Fe–Ni alloys. Finally we summarize the numerical results on the Curie temperature Slater–Pauling curves as well as the Slater–Pauling curves in 3d transition metal alloys. Let us start again from the local magnetic moment in the Hartree Fock approximation (8.6):   mi  = dω f (ω − μ) σρiσ (ω). (8.45) σ

The density of state ρiσ (ω) is given by the Green function Giiσ (z) as follows in the D-fold equivalent band model. ρiσ (ω) = −

D Im Giiσ (z). π

(8.46)

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Fig. 8.12 A cluster embedded on a lattice. The central atom is connected with z neighboring atoms via transfer integral t . Sj (Tj i ) denotes the contribution of all the paths which start from j and end at j (i) without returning to the cluster on the way

In the Hartree–Fock approximation, the Green function is given by (8.8):   Gij σ (z) = (z − H σ )−1 ij .

(8.47)

Here (H σ )ij = εiσ δij + tij (1 − δij ) is the one-electron matrix element for the Hartree–Fock Hamiltonian (8.4) and εiσ is the Hartree–Fock atomic potential on site i given by εiσ = εi0 + U˜ i ni /2 − J˜i mi σ/2. We consider the binary alloys and adopt the geometrical-mean model for the transfer integrals; tij = ri∗ t 0 rj . Here |ri | = (|tAA |/|tij0 |)1/2 (|ri | = (|tBB |/|tij0 |)1/2 ) when site i is occupied by atom A (B). The Green function ri∗ Gij (z)rj is then expressed as ri∗ Gij (z)rj =

−1   −1 L − t0 . ij

(8.48)

Here we have omitted the spin suffix for simplicity. The locator matrix L is defined by (L)ij = |ri |2 /(z − εiσ )δij . Equation (8.48) indicates that the Green function with the off-diagonal disorder is obtained from that of the diagonal disorder by replacing Gij (z) with ri∗ Gij (z)rj and (L)ij = δij /(z − εiσ ) with the locator (L)ij = |ri |2 /(z − εiσ )δij , respectively. Because of this we assume in the following the system with the diagonal disorder and use t instead of t 0 for simplicity. The Green function operator is expanded as G = (L−1 − t)−1 = L + LtG = L + LtL + LtLtG. Thus it is expressed as follows. Gij (z) = Li δij + Li tij Lj + +





Li tik Lk tkj Lj

k =i

Li tik Lk tkl Ll tlj Lj + · · · .

(8.49)

k =i l =k,j

This means that the Green function in the site representation is given by the contributions associated with all the paths which start from site i and end at site j . We consider a cluster consisting of the central atom on site 0 and the z neighboring atoms which are connected with the central site 0 via the nearest-neighbor transfer integral t. The cluster is embedded on a lattice as shown in Fig. 8.12. Using

8.3 Theory of Local Environment Effects in Magnetic Alloys

219

Fig. 8.13 The paths which start from j and end at 0 (left), and the paths which start from j and end at j  (right). The former corresponds to (8.51), the latter corresponds to (8.52)

the relation G = L + LtG, we obtain the Dyson equation as  t0j Gj 0 . G00 = L0 + L0

(8.50)

j

Classifying the terms in the locator expansion (8.49) according to the paths to the inside and outside of the cluster as shown in the left and the right of Fig. 8.13, we obtain the following equations  Gj 0 = Lj tj 0 G00 + Lj Sj Gj 0 + Lj Tj i Gi0 , (8.51) Gjj  = Lj δjj  + Lj tj 0 L0 t0j Gjj  + Lj + Lj





i =0,j

tj 0 L0 t0i Gij  + Lj Sj Gjj 

i =j

(8.52)

Tj i Gij  .

i =j

Here Sj (Tj i ) denotes all the paths which start from j and end at j (i) without returning to the cluster on the way. Now we consider the disordered alloys and replace the random potentials outside the cluster by a uniform effective potential Σ(z) called the coherent potential. This implies that the locators {Lj } outside the cluster have been replaced by an effective locator L (z). Accordingly, the self-energy Sj and the effective transfer integral Tj i are replaced by S and T , respectively. The Dyson equations (8.51) and (8.52) then reduce to the following ones.  Gi0 , (8.53) Gj 0 = Lj tG00 + Lj S Gj 0 + Lj T i =0,j

Gjj 



 = Lj δjj  + Lj L0 t 2 + S Gjj  + Lj L0 t 2 + T Gij  .

(8.54)

i =j

In the simplest approximation called the Bethe approximation, we neglect the effective transfer integral T assuming the Bethe lattice outside the cluster. We then

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obtain the off-diagonal Green function Gj 0 from (8.53). Substituting the expression into (8.50), we obtain the diagonal Green function as   G00 = L−1 0 − j

−1

t2

(8.55)

.

L−1 j −S

In the same way, we obtain the diagonal Green function Gjj on the boundary site from (8.54) without T as Gjj =

1 L−1 j

−S

+

1 (L−1 j

− S )2

t 2 G00 .

(8.56)

Here we used the relation (8.53). The effective self-energy S is obtained from (8.55) in which the random potentials on all the sites have been replaced by the coherent potential:  F = L −1 −

z t2 L −1 − S

−1 (8.57)

.

Here z denotes the number of the nearest neighbors (NN). The coherent Green function F is given by (8.14). In the following equations, we express the energy variable on the complex energy plane as ω + iδ when the number of the NN cannot be distinguished with the energy ‘z = ω + iδ’. Solving the above equation with respect to the effective self-energy S , we obtain S = L −1 −

z t2 . L −1 − F −1

(8.58)

The diagonal Green function (8.55) in the Bethe approximation is useful for taking into account the LEE on the magnetic moments because of its simplicity. But it does not include the effects of electron hoppings making a loop via outside of the cluster. The second approximation [168] takes into account both S and T . Solving (8.50) and (8.53) with respect to the diagonal Green function, we obtain  G00 = L−1 0 −

t2 −1 M −T

−1 (8.59)

.

Here M is defined by M=



−1 L−1 . n −S +T

(8.60)

n

The boundary-site Green function is obtained from (8.54) as Gjj =

1 L−1 j

−S +T

−

1

1 (L−1 j

−S +T

)2

M−

1 L0 t 2 +T

.

(8.61)

8.3 Theory of Local Environment Effects in Magnetic Alloys

221

Note that (8.59) and (8.61) reduce to (8.55) and (8.56), respectively when T is omitted. The effective self-energy S and the transfer integral T are obtained as functions of L from (8.59) and (8.61) in which all random potentials have been replaced by the coherent ones, i.e., from those in which G00 , Gjj , and Lj have been replaced by F (L ), F (L ), and L , respectively. The coherent potential or the effective locator L (ω + iδ) is determined by the self-consistent condition G00  = F,

(8.62)

Gjj  = F.

(8.63)

or

Here ∼ at the l.h.s. of the above equations means the configurational average. It is numerically shown that the self-consistent condition (8.63) at the shell boundary has better analytic property than the condition (8.62) at the central site [169]. The configuration of the cluster is specified by the number of surrounding atoms n of type α in the case of binary alloys. Here α denotes the type of the central atoms (i.e., A or B). The probability of finding n atoms of type α on the surrounding sites are given by the binomial distribution function.

 

n

z−n . Γ n, z, p αα = z!/n!(z − n)! p αα 1 − p αα

(8.64)

Here p αα is the probability of finding an atom α at a neighboring site of an atom α, and is given by Cowley’s atomic short-range order parameter τ as p αα = cα + (1 − cα )τ,

(8.65)

cα being the concentration of atom α. (Note that the parameter τ can vary from −cα /(1 − cα ) to 1). With use of the binomial distribution (8.64), the configurational average of the Green function G00 in (8.62) is given by G00  =

z 

Γ n, z, p αα G00 (ω + iδ, n).

(8.66)

n=0

In the Hartree–Fock self-consistent scheme, the potential εiσ changes as εiσ = εi0 + U˜ i ni /2− J˜i mi σ/2 depending on their environments. Thus, we have to solve self-consistently (8.45) and (8.63), as well as the equation of the charge potential  ni  = dω f (ω − μ) σ ρiσ (ω) for each environment. This is the Hartree–Fock cluster CPA theory. The theory can describe the LEE on the local magnetic moment self-consistently. We discuss in the following the LEE in Ni–Cu alloys as an example. Experimentally, the magnetization in Ni–Cu alloys monotonically decreases along the Slater–Pauling curve as shown in Fig. 8.1. The magnetic diffuse-scattering data of polarized neutron experiments however indicate a strong disturbance of Ni local

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Fig. 8.14 Magnetic moments as a function of Cu concentration in Ni–Cu alloys [173]. (a) The total magnetization and average Ni and Cu moments in the Hartree–Fock cluster CPA (solid lines) and in the Hartree–Fock CPA (dashed lines). The experimental values are represented by • [174],  [175],  [174], and  [176] for the total magnetization, and by  [176] for the average Ni moment. The average Cu moment is essentially zero. (b) The Ni atomic moments in various local environments (dashed lines) and the average Ni moment (solid line). Vertical bars represent the root-mean square deviations of the Ni moments from the average values. The local environment is specified by the number of Cu atoms on the NN shell

magnetic moments around 50 at% Cu [176]. Figure 8.14 shows the magnetization vs. concentration curves in Ni–Cu alloys and the local magnetic moment for Ni atom in various environments, which are calculated by the Hartree–Fock cluster CPA based on (8.59) and (8.61) [173]. In the case of Ni–Cu alloys, one has to take into account the hybridization of d electrons with sp electrons, whose effects are phenomenologically taken into account in the model. The critical concentration of the disappearance of the ferromagnetism is increased by the self-consistent local environment calculations as shown in Fig. 8.14(a); the LEE increase the ferromagnetic region. Figure 8.14(b) indicates that the Ni local moments (LM) are subject to strong local environment effects; the Ni LM surrounded by 12 Cu atoms almost lose the magnetic moment, while the Ni LM surrounded by 12 Ni atoms shows the full moment (see Fig. 8.15). These behav-

8.3 Theory of Local Environment Effects in Magnetic Alloys

223

Fig. 8.15 Schematic pictures showing the local magnetic moments (LM) of Ni in the environment n = 0 and n = 12 where n is the number of Ni nearest neighbors. In the former, the central LM of Ni disappears, while the LM recovers in the latter

Fig. 8.16 (a) The average DOS of Ni atom and (b) the local DOS of Ni atom in three local environments, which are specified by the number of Cu atoms on the nearest-neighbor shell at 30 at% Cu [173]. Long vertical lines are the Fermi level

iors are verified from the local densities of states (DOS) as shown in Fig. 8.16. The average DOS in Fig. 8.16(a) shows that the Ni70 Cu30 alloys are near the boundary between strong and weak ferromagnetism because the top of the up-spin DOS just touches on the Fermi level. The local DOS of a Ni atom with 12 Ni atoms on the NN shell has a sharp peak for both spins showing the strong magnetic moment augmented by surrounding Ni atoms. The DOS of a Ni atom with 12 Cu atoms on the NN shell has a Lorentzian type broad peak below the Fermi level so that Ni loses the magnetic moment. In Ni–Cu alloys all the local magnetic moments are in the same direction so that we can solve the Hartree–Fock self-consistent equations under a given direction of LMs. On the other hand, the Mn LMs in the Ni–Mn alloys are expected to change their direction depending on their environment because the coupling between Mn LMs is antiferromagnetic while the coupling between Mn and Ni LMs is ferromagnetic. It is not easy in this case to determine the direction of the LMs with use of the Hartree–Fock cluster CPA. The finite-temperature theory of the local environment effects [170–172] allows us to automatically determine the LM configuration, and describe the magnetic

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properties of alloys at finite temperatures. We start from the free energy (8.19) and the effective potential (8.23). Inserting the local magnetic field on site 0 and taking the derivative of the free energy F with respect to the field, we can derive the magnetic moment on the central site 0 (see (3.55)): m0  = ξ .

(8.67)

Here ξ denotes the exchange field variable on site 0. In the same way, we obtain the local charge on the central site 0 taking the derivative of F with respect to the atomic level on the same site as follows. n0  = n˜ 0 (ξ, ξ1 , ξ2 , . . .).

(8.68)

Here we defined the field variables ξ1 , ξ2 , . . . acting on the surrounding site 1, 2, . . . , respectively, bearing in mind that a cluster is centered at the site 0. The local charge n˜ 0 (ξ, ξ1 , ξ2 , . . .) at the central site 0 is defined by (8.21):   n˜ 0 (ξ, ξ1 , ξ2 , . . .) = dω f (ω) ρiσ (ω, ξ, ξ1 , ξ2 , . . .). (8.69) σ

The local densities of states ρiσ (ω, ξ, ξ1 , ξ2 , . . .) are calculated from the Green function for the Hamiltonian (8.24). Note that the average ∼ at the r.h.s. of (8.67) and (8.68) is a classical average with respect to the effective potential E(ξ, ξ1 , ξ2 , . . .). For any quantity A(ξ, ξ1 , ξ2 , . . .), it is defined by     N dξi A(ξ, ξ1 , ξ2 , . . .) e−βE(ξ,ξ1 ,ξ2 ,...) dξ   i . (8.70) A(ξ, ξ1 , ξ2 , . . .) =     N −βE(ξ,ξ1 ,ξ2 ,...) dξi e dξ i

We introduce here the effective medium Lσ into the energy (8.23) and express it using the relation (8.27) as    D E(ξ, ξ1 , ξ2 , . . .) = dω f (ω) Im tr ln(L − t) + E0 (ξ, ξ1 , ξ2 , . . .) π +

N 

Ei (ξi ) + E(ξ, ξ1 , ξ2 , . . .).

(8.71)

i=1

Here Ei (ξi ) are the single-site effective potentials (see (8.30)).   

 D −1 Ei (ξi ) = dω f (ω) Im ln 1 + L−1 iσ (z, ξ ) − Lσ (z) Fσ (z) π σ 1 1 (8.72) − U˜ i n˜ i (ξi )2 + J˜i ξi2 . 4 4 Here n˜ i (ξi ) is assumed to depend only on ξi . E0 (ξ, ξ1 , ξ2 , . . .) is defined by (8.72) in which the subscript i has been replaced by 0. Note that the second term E0 (ξ, ξ1 , ξ2 , . . .) at the r.h.s. of (8.71) may depend on the surrounding variable

8.3 Theory of Local Environment Effects in Magnetic Alloys

225

ξ1 , ξ2 , . . . via the charge potential U˜ 0 n˜ 0 (ξ, ξ1 , ξ2 , . . .)/2 which is determined selfconsistently. The last term of the r.h.s. of (8.71) is the nonlocal term defined by  

 D E(ξ, ξ1 , ξ2 , . . .) = dω f (ω) Im tr ln 1 − t˜F  . (8.73) π Expanding E with respect to the sites, we obtain the following expression in the lowest order.  E(ξ, ξ1 , ξ2 , . . .) = Φij (ξi , ξj ). (8.74) (i,j )

Here Φij (ξi , ξj ) is the pair energy defined by  D Φij (ξi , ξj ) = dω f (ω) π    × Im ln 1 − Fij σ (z)Fj iσ (z) t˜iσ (z) t˜j σ (z) ,

(8.75)

σ

and Fij σ is the off-diagonal component of the coherent Green function given by   Fij σ = (L − t)−1 ij σ , (8.76) t˜iσ is the diagonal component of the single-site t matrix (8.28): t˜iσ = −

−1 (L−1 iσ − Lσ )

−1 1 + (L−1 iσ − Lσ )Fσ

.

(8.77)

Substituting (8.71) into (8.70), we obtain an alternative form of the average as follows. A(ξ, ξ1 , ξ2 , . . .)     N pi (ξi ) dξi A(ξ, ξ1 , ξ2 , . . .) e−β[E0 (ξ,ξ1 ,ξ2 ,...)+E(ξ,ξ1 ,ξ2 ,...)] dξ =



i

dξ

  N

 pi (ξi ) dξi e

. −β[E0 (ξ,ξ1 ,ξ2 ,...)+E(ξ,ξ1 ,ξ2 ,...)]

i

(8.78)



Here pi (ξi ) = exp(−βEi (ξi ))/ dξi exp(−βEi (ξi )) is the single-site probability of finding the field ξi on site i. A simple way to reduce the integrals of the surrounding sites in (8.78) is to adopt the decoupling approximation which is correct up to the second moment in the mo 1/2 ment expansion; dξi pi (ξi )ξi2n+k ≈ xi2n ξi k0 where xi = ξi2 0 . Then for any function y(ξi ), we have   piν y(νxi ). (8.79) y(ξi )0 = pi (ξi )y(ξi ) dξi ≈ ν=±

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Magnetism of Disordered Alloys

Here piν = (1 + νξi 0 /xi )/2. It is interpreted as a probability that a local moment with an amplitude xi points up. Applying the decoupling approximation (8.79) successively in (8.78) and adopting the pair approximation (8.74), we obtain [177] A(ξ, ξ1 , ξ2 , . . .)     · · · A(ξ, s1 x1 , s2 x2 , . . .) e−βΨ (ξ,s1 x1 ,s2 x2 ,...) dξ s1 =± s1 =±



=

dξ



 

.

··· e

(8.80)

−βΨ (ξ,s1 x1 ,s2 x2 ,...)

s1 =± s1 =±

The effective potential Ψ (ξ, s1 x1 , s2 x2 , . . .) is expressed as  (a) Φ0j (ξ ) Ψ (ξ, s1 x1 , s2 x2 , . . .) = E0 (ξ, s1 x1 , s2 x2 , . . .) + j

 (e)  −1 −1 ξ 0 − + Kij si Φ0i (ξ ) + β tanh xi i =0

−



j =0,i

Jij si sj .

(8.81)

(i,j ) (a)

(e)

Here the pair interactions Φ0j (ξ ), Φ0i (ξ ), Kij , and Jij are defined as 1 1 (a) (e) Φ0j (ξ ) = Φij (ξ, νxj ), Φ0i (ξ ) = − νΦij (ξ, νxj ), 2 ν=± 2 ν=± Kij = −

1 λΦij (λxi , νxj ), 4 ν=± λ=±

Jij = −

(8.82)

1 λνΦij (λxi , νxj ). 4 ν=± λ=±

(8.83) As seen from (8.81), Jij is the exchange interaction between local magnetic moments (LM’s) in the decoupling approximation and reduces to the super-exchange interaction in the strong Coulomb interaction limit (see (1.86) and note that Jij = (e) 4Jij by definition when D = 1). Φ0i (ξ ) defined by (8.82) is the exchange energy (a) potential for the flexible central LM ξ . Φ0j (ξ ) is an atomic potential which changes the amplitude of LM ξ according to the surrounding atomic configuration. Kij is an effective nonlocal magnetic field induced by the polarization of the medium. Note that the latter vanishes in the paramagnetic state. It should be emphasized that the exchange couplings Jij provide us with the magnitude and sign of the couplings between the LMs in the itinerant electron system, thus they are useful for understanding the magnetic couplings between constituent magnetic atoms. The exchange couplings Jij are essentially the same as the Alexander–Anderson–Moriya interaction [178, 179], though the latter is based on the Anderson model (7.5). Equation (8.80) indicates that we can approximate the surrounding spins {ξi } with the Ising type of effective spins {si xi } with the amplitude {xi }. We consider

8.3 Theory of Local Environment Effects in Magnetic Alloys

227

now a cluster consisting of the central atom and the surrounding nearest-neighbor (NN) atoms, and calculate the central LM (8.67) on the basis of (8.80). The simplest approximation to the effective potential Ψ (ξ, s1 x1 , s2 x2 , . . .) is a molecular-field type in which the surrounding spins {si } are replaced by their averages mj /xj . In this case the effective potential is given by  (a)  (e) mj  Ψ (ξ ) = E0 (ξ ) + Φ0j (ξ ) − Φ0j (ξ ) . (8.84) xj j =0

j

In the disordered alloys, the intersite interactions are expected to be rather short range because of the damping effects due to random potentials. Equations (8.84) and (8.67) indicate that the central LM m0  is specified by the type of the central atom (α), the number of the NN (z), the types of the surrounding atoms ({γj }), and the surrounding LM ({mj }).  dξ ξ e−βΨ (ξ ) !

m0  = mα  {γj }, mj  =  . (8.85) dξ e−βΨ (ξ ) In order to determine the LMs in the cluster self-consistently, we introduce a distribution function gα (m) such that the probability of finding the LM of type α between m and m + dm is given by gα (m) dm. Then the probability that the surrounding atomic configuration {γj } is realized and each LM on site j with a type of atom γj has a value between mj and mj + dmj is given by $ $ [ zj =1 p αγj ][ zj =1 gγj (mj ) dmj ]. Here p αγ is the probability of finding atom γ at the neighboring site of atom α. It is given by (8.65) for γ = α and p αα = 1 − p αα for γ = α. The LM on the central atom α is then given by m0  = mα ({γj }, {mj }). Thus the probability that the central LM of the type of atom α is between M and M + M is given by  z  z     αγj gα (M)M = p gγj (mj ) dmj . (8.86) {γj } M≤mα ≤M+M j =1

j =1



Inserting the identity dM  δ(M  − mα ) and making use of the binomial distribution function (8.64), the above equation is written as follows.  z 



αα δ M − mα  Γ n, z, p gα (M) = n=0

×

 n  i=1

 gα (mi ) dmi

z 

 gα (mj ) dmj .

(8.87)

j =n+1

This is known as the distribution function method [180, 181]. The self-consistent integral equation (8.87) for gα (M) is not easy to solve for large z (e.g., 8 for the bcc, 12 for the fcc alloys). A way to solve the equation

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Magnetism of Disordered Alloys

is to apply the decoupling approximation of the type of (8.79) to the distribution functions {gγ (mj )} at the r.h.s. of (8.87).  n   k m2n+k gγ (m) dm ≈ mγ 2 c mγ  c (k = 0, 1), (8.88) where [∼]c denotes the configurational average. This implies that we can make the following approximation for any function f (m) at the r.h.s. of (8.87).    1  1/2

[mγ ]c f (m)gγ (m) dm ≈ f ν mγ 2 c . (8.89) 1+ν 1/2 2 2 [mγ  ]c ν=± After making the decoupling approximation, we substitute the  approximate distribution gα(M) (i.e., (8.87)) into the equations [mα ]c = Mgα (M) dM and [mα 2 ]c = M 2 gα (M) dM, and obtain the self-consistent equations for [mα ]c and [mα 2 ]c as follows. 

  z

[mα n ]c [mα ]c αα = . Γ n, z, p [mα 2 ]c [mα 2n ]c

(8.90)

n=0

The magnetic moments [mα kn ]c (k = 1, 2) at the r.h.s. of the above equations are the LM’s for a given environment n. They are given by n  z−n    k mα n c = Γ (l, n, qα+ )Γ (m, z − n, qα+ )ξα knlm .

(8.91)

l=0 m=0

Here qα± = (1 ± [mα ]c /([mα 2 ]c )1/2 )/2 is the probability that the LM of type α points up or down. ξα nlm is the LM of an atom of type α at the central site when l of the LMs among the surrounding n atoms of type α point up, and in addition, m LMs of the remaining z − n atoms of type α also point up:  (8.92) ξα nlm = pαnkl (ξ )ξ dξ, pαnlm (ξ ) = 

e−βΨαnlm (ξ ) , dξ e−βΨαnlm (ξ )

(8.93) 1/2

(a) (a) Ψαnlm (ξ ) = Eα (ξ ) + n Φαα (ξ ) + (z − n) Φαα (ξ ) − (2l

(e) − n) Φαα (ξ )

[mα 2 ]c xα

1/2

(e)

− (2m − z + n) Φαα (ξ )

[mα 2 ]c . xα

(8.94)

Equation (8.90) determines [mα ]c and [mα 2 ]c self-consistently when the (a) (e) medium Lσ−1 is given, because Eα (ξ ), Φαγ (ξ ), and Φαγ (ξ ) are the functionals

8.3 Theory of Local Environment Effects in Magnetic Alloys

229

of the medium Lσ−1 . We determine in principle the latter from the CPA equation (8.34) or (8.36). However, in the present case, the thermal average of the single-site Green function depends on the surrounding environment {γj } and {mj } via the effective potential (8.84), so that the CPA equation is expressed as    −1 −1 −1 cα [L−1 (8.95) ασ (z, ξ ) − Lσ (z) + Fσ (z)]  c = Fσ (z). α

Here the average ∼ should be taken with respect to the effective potential (8.84). Applying again the decoupling approximation of the type (8.79) for the average [∼]c , we obtain a simplified CPA equation as follows.    1 −1 [ξα ]c  −1  2 1/2

Lασ z, ν ξα  c − Lσ−1 (z) + Fσ−1 (z) 1+ν cα 1/2 2 2 [ξα ]c α ν=± = Fσ (z).

(8.96)

Here [ξαk ]c (k = 1, 2) is defined by z n  z−n   k 

 ξα  c = Γ n, z, p αα Γ (l, n, qα+ )Γ (m, z − n, qα+ )ξαk nlm , (8.97) n=0

l=0 m=0

ξαk nlm =

 ξ k pαnlm (ξ ) dξ.

(8.98)

Equations (8.90) and (8.96) determine [mα ]c , [mα 2 ]c , and Lσ−1 self-consistently. It is worth to mention that the self-consistent equations (8.90) contain the spinglass solution, [mα ]c = 0 and [mα 2 ]c = 0 (see (1.104)). To see the fact, we consider the case [mα ]c = 0, and write the second equation of (8.90) for vα = 1/2 [mα 2 ]c /xα as follows after making the decoupling approximation (8.79) for the central-site variable ξ . vα2 =

z  n  z−n 

Γ (n, z, cα )Γ (l, n, 1/2)Γ (m, z − n, 1/2)

n=0 l=0 m=0

  × tanh2 β(2k − n)Jαα vα + β(2l − z + n)Jαα vα .

(8.99)

Here we assumed the complete disorder p αα = cα . Expanding the r.h.s. of (8.99) with respect to vα and assuming that Jαγ is independent of temperature, we obtain the transition temperature Tg at which vα vanishes. 1 5 2 2 Tg2 = z cA JAA + cB JBB + 2

'

6

2 − c J 2 2 + 4c c J 4 . (8.100) cA JAA B BB A B AB

The above expression of Tg reduces to the spin-glass temperature (7.46) for the insulator system in the dilute limit.

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Fig. 8.17 Various local magnetic moments (LM) in Ni–Mn alloys as a function of Ni concentration at 150 K [172]. Solid curves are calculated results of [mMn ]c (upper curve) and [mNi ]c (lower curve). Corresponding experimental values at T = 4.2 K (room temperatures) are denoted by  and  ( and ) [187], respectively. The dotted curve represents the calculated magnetization. The experimental values are shown by ◦ (4.2 K) and • (room temperatures) [187, 188]. 1/2 The root mean square values of the thermal average of Mn and Ni LM’s [mα 2 ]c are shown by dot-dashed curves

The magnetic properties of Ni–Mn alloys cannot be explained by the singlesite approximation as mentioned before. The magnetization vs. concentration curve shows a deviation from the Slater–Pauling curve as shown in Fig. 8.1. The system is characterized by the strong atomic short range effects on magnetic moment [182–184], and the existence of spin glasses [185, 186]. Finite temperature calculations based on the LEE theory mentioned above allows us to explain these properties of Ni–Mn alloys. Calculated magnetic moments vs. concentration curves are presented in Fig. 8.17. The magnetization curve shows a maximum at 10 at% Mn in agreement with the experimental data. The deviation from the linear Slater–Pauling curve is caused by the rapid decrease of average Mn local magnetic moment (LM) with increasing Mn concentration. The Mn atoms have well-defined LM with a large amplitude more than 3μB , and the magnetic couplings in Ni–Mn alloys are verified to be JMnMn < 0, |JMnMn |  JNiNi > 0, and JNiMn > 0. Thus it is expected that the rapid decrease of average Mn LM is accompanied by the reversal of the Mn LM with the increase in Mn concentration. Figure 8.18 shows the calculated concentration dependences of Mn and Ni LMs in various local environments. At low Mn concentrations, Mn LM are parallel to the magnetization when the number of surrounding Mn LM n is less than n = 3, while they are antiparallel when n are equal to or larger than 3. Such behavior explains the concentration dependence of the magnetization by taking into account the binomial

8.3 Theory of Local Environment Effects in Magnetic Alloys

231

Fig. 8.18 Concentration dependences of the LM in various environments, [mα n ]c [172]. The solid (dashed) curves show the Mn (Ni) LM. Integers n in the figure show the coordination number of Mn

distribution via (8.90). Beyond 10 at% Mn, the Mn LM with critical environments n = 2, 3, 4 rapidly reduce their magnitudes with decreasing average polarization. This is because the Mn LM with critical environments feel a weak molecular field. Due to the same reason, they lose rapidly their magnetic moment with increasing temperature. The Ni LM, on the other hand, do not change their directions irrespective of their local environments as seen in Fig. 8.18, so that the average Ni LM monotonically decreases with increasing concentration as shown in Fig. 8.17. Note that [mα 2 ]c remain beyond 27 at% Mn after the disappearance of the ferromagnetism, indicating the existence of the spin glass state in agreement with the experimental data [185, 186]. The Ni–Mn alloy forms the Cu3 Au ordered alloy at 25 at% Mn. Experimentally the degree of order is controlled by quenching. The degree of atomic order is specified by the atomic short-range order (ASRO) parameter τ defined by (8.65); the Cu3 Au ordered state is characterized by τ = −cMn /cNi = −0.33, while the complete disorder is characterized by τ = 0. It is found that the magnetization is strongly influenced by the ASRO at 25 at% Mn; it increases from 0.1μB to 1.0μB when the ASRO decreases from τ = 0 to τ = −0.33 [184]. Figure 8.19 shows the local densities of states (LDOS) in various environments at 25 at% Mn for complete disorder. The up-spin electrons on Mn atoms with no Mn NN hybridize with surrounding electrons on Ni atoms with the same spin, while the down-spin electrons on Mn atoms are localized above the Fermi level. The Mn LMs with 12 Mn NN are subject to the ferro- and antiferro-magnetic molecular fields depending on the surrounding Mn configurations in direction, so that the LDOS hardly depend on the spin component. On the other hand, the Ni LDOS with no Mn

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Magnetism of Disordered Alloys

Fig. 8.19 Local densities of states (LDOS) in various environment at 25 at% Mn and 150 K [172]. Solid curves show the LDOS in the environment with no nearest neighbor (NN) Mn atoms. Dashed curves, LDOS with n = 6 Mn NN. Dotted curves, LDOS with n = 12 Mn NN

NN is well polarized due to the Ni cluster. With increasing Mn NN, the polarization of Ni LM decreases and the peak of the DOS sinks below the Fermi level. In the Ni LDOS with 12 Mn NN, we find considerable DOS above the Fermi level due to the hybridization with the electrons on Mn sites. Calculated magnetization vs. temperature curves at 25 at% Mn are presented in Fig. 8.20 for various ASRO. At low temperatures, it is verified that there are the low-spin state and high-spin state depending on the ASRO, and the first-order transition from the former to the latter takes place between τ = −0.18 and τ = −0195 due to the change in magnetic couplings with decreasing the number of surrounding Mn LM. The high LM states rapidly decrease with increasing temperature because

8.3 Theory of Local Environment Effects in Magnetic Alloys

233

Fig. 8.20 (a) Calculated magnetization vs. temperature curves for various ASRO parameters τ at 25 at% Mn [172]. (b) Experimental magnetization vs. temperature curves for various ASRO parameters τ at 23.7 at% Mn [184]

of larger magnetic entropy. When the number of Mn NN decreases with decreasing τ , the Ni–Mn ferromagnetic couplings JNiMn become dominant, so that high magnetization and high TC are realized at τ = −0.33. As a second example showing strong LEE, we consider the Fe–Ni alloys. The alloys show a rapid deviation from the Slater–Pauling curve at c∗ = 66 at% Fe as shown in Fig. 8.1. The ferromagnetic instability takes place when the up-spin band touches the Fermi level, as discussed in the last section. The ferromagnetic instability with strong LEE causes various anomalies such as the broad internal-field distribution in Möbauer experiment [189], the downward deviation of the magnetization vs. temperature curve from the Brillouin curve [190], the large magneto-volume ef-

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Magnetism of Disordered Alloys

Fig. 8.21 Concentration dependence of various local moments (LM) in Fec Ni1−c alloys calculated for the bcc (c ≥ 0.65) and fcc (0 ≤ c ≤ 1) structures at T = 150 K [192]. Dotted curves: average magnetizations, solid curves: partial magnetization [mα ]c , dashed curves: 1/2 amplitudes of LM [m2α ]c , dot-dashed curves: 1/2 [mα 2 ]c . Experimental magnetizations are shown by ◦ [190, 193, 194]. Experimental LM [mα ]c at 4.2 K are shown by  (α = fcc Fe),  (α = fcc Ni) [155, 156, 193],  (α = bcc Fe), and  (α = bcc Ni) [193]

fects called the Invar effects [191], and the appearance of spin glasses (SG) after the disappearance of ferromagnetism [163]. The finite-temperature theory of the LEE explains these characteristics of Fe–Ni alloys. Figure 8.21 shows the concentration dependence of various magnetic moments calculated by the theory of LEE [192]. Calculated magnetization linearly increases with increasing Fe concentration and starts to decrease rapidly at 66 at% Fe, where the top of the up-spin band touches the Fermi level. Calculated magnetic moments 1/2 [mα 2 ]c remain even after the disappearance of ferromagnetism, showing the existence of the SG. Experimentally, the SG are found in the fcc (Fec Ni1−c )92 C8 alloys, because the Fe–Ni alloys cause the structural phase transition to the bcc after the disappearance of ferromagnetism beyond 70 at% Fe [163]. The strong LEE found in Fe–Ni alloys originate in a nonlinear magnetic coupling between Fe LMs. Figure 8.22(a) shows various atomic and exchange pair energies of the central LM of type α when the neighboring LM of type γ with amplitude xγ points up. If the exchange interaction Φ(ξi , ξj ) follows the bilinear form like (e) the Heisenberg model (−Jij ξi · ξj ), exchange energy potential −Φαγ (ξ ) should be linear with respect to ξ and there is no anomaly in the vicinity of ferromagnetic (e) instability. However, the exchange potential −ΦFeFe (ξ ) in the fcc structure shows an S-shape curve. It implies that Fe LMs with the average amplitude (ξ 2 1/2 ) less than about 1.7μB couple antiferromagnetically to the neighboring Fe LMs, while the Fe LMs with the amplitude more than 1.7μB couple ferromagnetically to the (a) (a) surrounding Fe LMs. Since the pair-energy functionals ΦFeFe (ξ ) and ΦFeNi (ξ ) show downward and upward convex curves, respectively, the amplitude of the central LM varies from 2.6μB to 1.5μB with the increasing number of Fe NN. This means that the Fe LMs with a small number of Fe NN show ferromagnetic coupling to

8.3 Theory of Local Environment Effects in Magnetic Alloys

235

(a) (e) Fig. 8.22 Pair-energy potentials Φαγ (ξ ) and −Φαγ (ξ ) for fcc (upper) and bcc (lower) Fe65 Ni35 alloys calculated at 900 K [192]. The notation αγ (at) (αγ (ex)) indicates the atomic (exchange) coupling between atom α and atom γ

the neighboring Fe LM, while the Fe LMs with a large number of Fe NN show the antiferromagnetic coupling to the neighboring Fe LM’s (see Fig. 8.23). The (e) antiferromagnetic coupling of ΦFeFe (ξ ) does not appear for any value of ξ in the bcc lattice as shown in Fig. 8.22(b). The Fe LMs in Fe–Ni alloys decrease their amplitudes (ξ 2 1/2 ) with increasing (a) Fe concentration due to the downward-convex coupling ΦFeFe (ξ ) and the upward(a) convex coupling ΦFeNi (ξ ). Near the critical concentration of ferromagnetic instability, amplitudes of LM with more than 10 Fe NN become less than 1.7μB and the Fe atoms with such local environments reverse their LM, leading to the ferromagnetic instability.

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Magnetism of Disordered Alloys

Fig. 8.23 Nonlinear coupling between Fe LM’s in the fcc Fe–Ni alloys. Fe LM’s with less than ten Fe NN have large amplitudes ξ 2 1/2 more than 1.7μB and therefore show the ferromagnetic coupling, but Fe LM’s with more than ten Fe NN have small amplitudes less than 1.7μB , thus the antiferromagnetic coupling

Fig. 8.24 Distribution functions gFe (M) of Fe LM in Fe–Ni alloys at 150 K for various Fe concentrations [192]

The nonlinear coupling between Fe LMs and its LEE yield a broad distribution of Fe LM near the critical concentration as shown in Fig. 8.24. Note that even after the disappearance of ferromagnetism, a broad LM distribution remains. It implies the existence of the SG specified by [mα ]c = 0 and [mα 2 ]c = 0. Strong disturbance of Fe LM due to the LEE is also caused by thermal excitations. As shown in Fig. 8.23, the Fe LMs with more than 10 Fe NN decrease their amplitudes with increasing temperature near the critical concentration, and thus cause the reversal of their LM. Figure 8.25 shows this behavior (see the distribution at 0.53 TC ). With further increase of temperature, the distribution shrinks and merges into zero value at TC .

8.3 Theory of Local Environment Effects in Magnetic Alloys

237

Fig. 8.25 Temperature dependence of the distribution functions for Fe LM at 62.5 at% Fe in Fe–Ni alloys [192]

We can calculate the internal field distribution of 57 Fe in Mössbauer experiments with use of an empirical expression Hi = ai mi  + zj =1 bij mj  [196]. Here ai and bij are constants. By making use of the distribution functions gα (M), we obtain the distribution of the internal field on atom α as follows.  z 



δ H − Hα n, z, {mi } Pα (H ) = Γ n, z, p αα n=0

 ×

n 

 gα (mi ) dmi

z 

 gα (mj ) dmj .

j =n+1

i=1

(8.101) Here Hα (n, z, {mi }) is the internal field under a given configuration, and is expressed by n z  



mi + bαα mj . Hα n, z, {mi } = aα ξα  n, z, {mi } + bαα i=1

(8.102)

j =n+1

Figure 8.26 shows the calculated temperature dependence of the internal-field distribution function PFe (H ) at 62.5 at% Fe. With increasing temperature, the distribution rapidly broadens in accordance with the broadening of gFe (M) at the same temperature. The results agree with the experimental data depicted in the inset [189]. A peak at H ≈ 0.7μB in the distribution functions for 0.53 TC and 0.74 TC originates in the Fe LM with 9 or 10 Fe NN. Near the critical concentration of ferromagnetic instability, the magnetization vs. temperature curves deviate downward from the Brillouin curve experimentally [190]. Both the single-site theory and the theory of LEE can explain the fact

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Magnetism of Disordered Alloys

Fig. 8.26 Temperature dependence of the internal-field distribution functions for 57 Fe at 62.5 at% Fe [192]. The inset shows the experimental results at 65 at% Fe [189]

Fig. 8.27 Reduced magnetization curves near the critical concentration of the ferromagnetic instability [171]. The solid (dot-dashed) curve is the result of the finite-temperature theory of the LEE (the SSA) at 50 at% Fe. The dashed curve is the S = 1/2 Brillouin curve. The open circles are the experimental data at 67 at% Fe

as shown in Fig. 8.27. The theory of LEE yields the smoother curve being close to the experimental data because the fluctuations of the surrounding configurations are favorable to the second order transition. Note that the other type of excitations also becomes important near the critical concentration of ferromagnetic instability. The low-energy spin wave excitations and the electron excitations coupled to the spin waves may also contribute to the smooth magnetization vs. temperature curve at low temperatures. The calculated magnetic phase diagram of Fe–Ni alloys is summarized in Fig. 8.28. The result explains the experimental data semi-quantitatively as shown

8.3 Theory of Local Environment Effects in Magnetic Alloys

239

Fig. 8.28 Calculated magnetic phase diagram showing the ferromagnetic (F), spin-glass (SG), and paramagnetic (P) states [192]. The Curie temperatures for the bcc structure are shown by the dashed curve. The inset shows the experimental result [146]

Fig. 8.29 Theory (dashed curves) vs. experiment (solid curves) for the Slater–Pauling curves. The calculations are performed at 150 K [195]

in the inset. The Curie temperature of Fe–Ni alloys shows a maximum with increasing Fe concentration because the ferromagnetic coupling JFeNi is stronger than JNiNi and the antiferromagnetic coupling JFeFe (< 0) is weaker than the others in magnitude. Magnetization vs. concentration curves in 3d transition metal alloys, which are calculated by the theory of LEE and the d band model, are summarized in Fig. 8.29. As discussed in Sect. 8.1, the straight line with −45◦ slope at the r.h.s. of the Slater– Pauling curves is explained by the increase of holes in the down-spin band of the strong ferromagnet with decreasing average d electron number n. The Fermi level

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Magnetism of Disordered Alloys

reaches the top of the up-spin band around n = 7.5. When the d electron number becomes smaller than around 7.5, holes are created in the up-spin band. In Fe–V alloys the V local moments are antiparallel to the magnetization. The magnitudes of both Fe and V LMs decrease with increasing V concentration. Although the concentration dependence of the magnetization curve in Fe–V alloys is explained by a simple dilution picture, it does not necessarily mean that the LEE are not important. Since the number of holes in the d bands increases with adding V atoms, Fe LMs become more flexible, so that they show the broad distribution of LMs. The distribution is caused by the LEE, i.e., the strong fluctuations in amplitudes due to the fluctuation of the surrounding atomic configuration. These LEE explain the broad distribution of the internal field seen by 57 Fe in Mössbauer experiments [197]. The Fe–Cr alloys show the behaviors similar to that of the Fe–V alloys with strong LEE. The theory explains the concentration dependence of magnetization. It also explains the concentration and temperature dependences of the distribution of the internal field in Mössbauer experiments. The Cr-rich Fe–Cr alloys show the antiferromagnetic state at less than 19 at% Fe. The theory of LEE is extended to the antiferromagnetic case under the assumption of the two sublattices [198]. In this case, we introduce two kinds of effective media, Lσ(+) on the (+) sublattice (−) and Lσ on the (−) sublattice. Accordingly, we have the distribution functions (+) (−) gα (M) and gα (M) on the (+) and (−) sublattices, respectively. Because of the (−) (+) (−) (+) symmetric relations Lσ = L−σ and gα (M) = gα (−M), we have the same type of integral equation (8.87) in which the distribution functions {gγi (mi )} at the r.h.s. of the equation have been replaced by {gγi (−mi )}. The upward deviation of the calculated curve in Ni–Cu alloys originates in the neglect of the sd hybridization. The hybridization delocalizes Ni LMs therefore yields smaller magnetization in Cu-rich Ni alloys, so that the linear-concentration dependence is realized (see Fig. 8.14). The branch in Ni–Cr alloys has been explained by the formation of the virtual-bound state in the up-spin band above the Fermi level when the Cr atoms are dissolved in the host Ni (see Fig. 8.7). The deviation from the straight line in Ni–Mn alloys has been shown to be caused by the reversal of Mn LMs with more than 3 Mn NN atoms with increasing Mn concentration (see Fig. 8.17). It is also worth to remind that the SG state appears after the disappearance of ferromagnetism, in agreement with the experiment [185], since the ferromagnetic exchange couplings between Ni LMs and between Ni and Mn LMs compete with the antiferromagnetic couplings between Mn LMs. Rapid but continuous decrease of the curve in Fe–Ni alloys is not described by the single-site theory. The second-order transition with increasing Fe concentration is caused by nonlinear magnetic couplings between Fe LMs and the LEE on both the magnetic moments and their amplitudes of Fe LMs, as we have discussed before (see Fig. 8.21). The nonlinear couplings and the LEE on Fe LMs also describe the SG in Fe–Ni alloys [163] after the disappearance of ferromagnetism. The same type of strong LEE is found in fcc Co–Fe alloys [199], though the alloys change the structure from the fcc to the bcc at 30 at% Fe. According to the theoretical calculations, the fcc Co–Fe alloys show the ferromagnetic instability around 78 at% Fe and

8.3 Theory of Local Environment Effects in Magnetic Alloys

241

Fig. 8.30 Theory (dashed curves) vs. experiment (solid curves) for the Curie-temperature Slater– Pauling curves [195]. Calculated results are renormalized at the pure metals

the SG behavior after the disappearance of ferromagnetism due to the same reason as in the case of Fe–Ni alloys. These behaviors are verified in the fcc Fe–Co particle system precipitated in a Cu matrix [200]. It is well-known that the Curie temperatures (TC ) in 3d transition metals show the curves being similar to the Slater–Pauling curves. The Curie-temperature Slater– Pauling curves calculated by the theory of the LEE are presented in Fig. 8.30 with the experimental data. Since TC are determined by thermal excitations, the curves are not directly related to the ground-state magnetizations. It should be noted that theoretical values of TC remain qualitative or semi-quantitative due to various approximations, so the curves in Fig. 8.30 are renormalized by TC for one of the constituent metals. Calculated Curie-temperature Slater–Pauling curves explain overall features of the experimental data. In qualitative discussions on the concentration dependence of TC , the effective exchange interactions {Jαγ } defined by (8.83) are useful. In the case of Fe–V alloys, TC shows a maximum with increasing V concentration. In this system, calculated exchange interactions show that JFeFe ∼ |JFeV |  |JVV | > 0, and JFeV , JVV < 0 in the bcc pure Fe. With increasing V concentration, JFeFe rapidly increases due to alloying and shows a maximum at 25 at% V, while |JFeV | monotonically decreases. These concentration dependences of magnetic couplings cause a maximum in TC around 20 at% V in agreement with the experiment. The same type of alloying effect on TC is seen in the bcc Fe–Ni alloys. There the exchange coupling JFeFe rapidly decreases up to 10 at% Ni with increasing Ni concentration. The effects decrease TC in spite of the fact that JFeNi > JFeFe > 0. On the other hand this alloying effect is not found in the bcc Fe–Co alloys. There the magnetic couplings satisfy an inequality JCoCo > JFeCo > JFeFe > 0 and JFeFe gradually increases with increasing Co concentration, so that TC in the bcc Fe–Co alloys monotonically increases.

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Magnetism of Disordered Alloys

The maximum in the curve for fcc Fe–Ni alloys and the curves in Fe–Co–Ni alloys are explained basically by a rigid band theory with thermal spin fluctuations. More detailed analyses indicate that the Fe–Ni pairs JFeNi (> JNiNi ) increase TC first and the Fe–Fe pairs JFeFe (< 0) decrease TC next with increasing Fe concentration. The Curie temperature in Ni–Mn alloys monotonically decreases with increasing Mn concentration, while the magnetization vs. concentration curve shows a maximum. The former is explained by the fact that JNiNi > JNiMn > 0 up to 10 at% Mn. The latter is explained by the fact that the small Ni LMs are replaced first by the large Mn LMs parallel to the magnetization, but the Mn LMs become antiparallel to the magnetization when the number of neighboring Mn atoms become larger than 3, as discussed before.

8.4 Computer Simulations for Disordered Magnetic Alloys The finite-temperature theory of local environment effects (LEE) self-consistently determines the distribution of the local moments (LM) of constituent atoms in disordered alloys going beyond the single-site approximation, and allows us to calculate their temperature dependences. The method however relies on the distribution of LM, and therefore does not provide us with details on the magnetic structure in real space. In addition, it is not easy to take into account the effects of inter-site correlations going beyond the nearest-neighbor approximation because the number of integrals rapidly increases with increasing the correlation length in the integral equation for distribution functions. The molecular dynamics (MD) approach presented in Sect. 6.3 allows us to clarify the spatial structure of magnetic moments and to take into account the second, third, and further distant neighbor correlations in disordered alloys. In this section we describe the MD approach to the disordered alloys in greater details and present its application to the Fe–Cr alloys [108, 109, 201]. In the MD approach, we calculate the thermal average of LMs mi  by means of the time average given by (6.63). The spin dynamics leading to the thermal average (6.60) is obtained by solving the equations of motion (6.64)–(6.66). At each time step, we calculate the magnetic forces −∂Ψ (ξ )/∂ξiα . The latter is obtained from (6.67) via the temporal LM mi 0 as −∂Ψ (ξ )/∂ξiα = (J˜/2)(miα 0 − ξiα ) − 2T ξiα /ξi2 , where the average (∼)0 is taken with respect to the Hamiltonian (6.57). For simplicity we consider here the d band model Hamiltonian obtained from (6.57), and assume the alloys with off-diagonal disorders described by the geometrical mean model: 0 timj m = rα∗ timj m rγ ,

(8.103)

where α (γ ) denotes the type of atom A or B on sites i (j ) (see (8.16)). The random atomic configuration for a given concentration cα and the ASRO parameter τα is made on the computer as follows. Let us consider a cubic unit cell consisting of a large number of N atoms with periodic boundary condition. The

8.4 Computer Simulations for Disordered Magnetic Alloys

243

ASRO parameter is connected with the probability of finding an atom α on the nearest-neighbor of atom α as p αα = cα + (1 − cα )τα (see (8.65)). We first create an A or B atom on each site of the MD unit cell with the probability cA or cB using random numbers. Next, we replace an A atom with a B atom (or a B atom with an A atom) at randomly chosen sites until the given concentration cA is realized.  We then calculate the probability p˜ AA = i∈A NAA (i)/NA z, i.e., the probability of finding another A atom at the neighboring site of an A atom. Here NAA (i) denotes the number of A atoms at the nearest-neighbor sites of the A atom at site i, z is the number of nearest-neighbor sites, and NA is the total number of A atoms in the MD unit cell. If the probability p˜ AA is smaller (larger) than p AA , we exchange the A atom with the B atom in a randomly chosen A–B NN pair, and accept the exchange if the new p˜ AA is larger (smaller) than the old one. We obtain the requested alloy by repeating the procedure until |p˜ AA − p AA | ≤ ε(= 2/NA z) is satisfied. The magnetic moment on site i in the magnetic force is expressed by means of the Green function G(z) as follows.  miα 0 =

dω f (ω)

Gimσ j m σ  (z) =

(−)  Im σα G(z) imσ imσ , π mσ



−1  z − H (ξ ) . imσ j m σ 

(8.104) (8.105)

Here H (ξ ) is the one-electron Hamiltonian matrix of (6.57) in which the matrix elements for the sp electrons have been dropped. When we calculate the Green function on site i in (8.104), we consider a MD unit cell in which site i is centered. We surround the MD unit cell with 27 MD unit cells in which all atomic levels or locators have been replaced by an effective medium (see Fig. 6.6). As we have discussed in Sect. 6.3, the Green functions in (8.104) are expressed as follows by using the new basis representations which diagonalize the Pauli spin matrices σα (α = x, y, z). 

(σx G)imσ imσ = Gim1im1 − Gim2im2 ,

(8.106)

(σy G)imσ imσ = Gim3im3 − Gim4im4 ,

(8.107)

 (σz G)imσ imσ = Gim↑im↑ − Gim↓im↓ .

(8.108)

σ

 σ

σ

Here the√local basis functions at the r.h.s. √ are defined by |im1 = (|im √ ↑ + |im ↓)/ 2, |im2 = (|im ↑√− |im ↓)/ 2, |im3 = (|im ↑ + i|im ↓)/ 2, and |im4 = (|im ↑ − i|im ↓)/ 2. The diagonal Green function Gimαimα (α = 1–4, ↑, ↓) can be calculated by using the recursion method as follows (Appendix G).

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Gimα imα (z, ξ ) 1

=

.

|b1imα (ξ )|2

z − a1imα (ξ ) −

|b2imα (ξ )|2

z − a2imα (ξ ) −

···

··· ··· −

|bl−1imα (ξ )|2 z − alimα (ξ ) − Tlimα (z, ξ ) (8.109)

Here animα (ξ ) and bnimα (ξ ) are the recursion coefficients of the n-th order. Tlimα (z, ξ ) is called the terminator. When the recursion coefficients begin to contain the matrix elements outside the MD unit cell at the l-th level, we approximate the terminator Tliνα (z, ξ ) with an effective terminator for the coherent Green function [(L −1 − t)−1 ]imα imα /|ri |2 : Tlimα (z, ξ ) ≈ |rα |2 Tlmα . Here Tlmα is the terminator of the lth level obtained from the coherent Green function. In Sect. 8.2, we have discussed the ferromagnetic Fex Cr100−x alloys (20  x ≤ 100) in the ferromagnetic region within the single-site approximation, and in the last section we briefly discussed the LEE of Fe–Cr alloys in the same region. The Fe–Cr alloys are reported to show the antiferromagnetism (AF) in the Cr-rich region (x  5 at% Fe), while in the Fe-rich region (90 at% Fe  x), they show a simple ferromagnetism (F) with Cr local magnetic moments (LM) being antiparallel to the bulk magnetization. The actual Fe–Cr alloys however show more complex features due to long-range competing magnetic interactions. In fact, the system shows in the concentrated regions complex magnetic structures and its phase diagram has not been established yet. In particular, in the range 10 at% Fe  x  30 at% Fe where the phase boundary of the F in Fe-rich region encounters with that of the Cr-rich AF, two kinds of phase diagrams are proposed experimentally: the phase diagram proposed by Loegel [202] and Rode et al. [203] in which the F and the AF overlap (coexistence of F and AF) at zero temperature, and the phase diagram proposed by Burke et al. [204, 205] which shows up the existence of the spin-glass (SG) like phases between the F and the AF. Also, in the region above the F phase boundary (20 at% Fe  x  25 at% Fe), temperature dependent complex magnetic structures are suggested in neutron [206] and Mössbauer experiments [207, 208]. Furthermore, the real microscopic magnetic structures of Fe–Cr alloys in the range 30 at% Fe  x  70 at% Fe are expected to be rather complex because of the competition between the ferro- and antiferro-magnetic couplings, although the composition dependence of the average magnetic moment in the Fe-rich region was explained by a constant number of holes in the down spin band in the rigid band picture (see Sect. 8.1). In what follows, we elucidate the complex magnetic structure obtained by the MD method [201] to see how much information one can obtain from a computer simulation. The magnetic structures based on the molecular dynamics method are presented in Figs. 8.31–8.36. In the computer simulations, N = 250 atoms are put in a MD

8.4 Computer Simulations for Disordered Magnetic Alloys

245

Fig. 8.31 Magnetic structure of Fe80 Cr20 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms. The Cr clusters are shown by the nearest-neighbor Cr networks which are represented by lines and dots on the Cr atoms

unit cell which corresponds to the 5 × 5 × 5 bcc lattice, and the temperature has been kept at 25 K. When the Cr atoms are added to the ferromagnetic Fe, the calculated Cr LMs are aligned to be antiparallel to the bulk magnetization; the number of the down spin states on Cr atoms below the Fermi level becomes larger than that of the up spin states due to more hybridization of former electrons with the same spin bands of the host Fe. The impurity states of Cr atoms continue to exist until 90 at% Fe, where small clusters consisting of the nearest-neighbor (NN) Cr bonds begin to appear in the ferromagnetic Fe matrix. At 80 at% Fe, about two thirds of all Cr atoms belong to small clusters and the others are still isolated as shown in Fig. 8.31. The isolated Cr LMs are antiparallel to the bulk magnetization, while the Cr LMs in the clusters are declined due to the AF couplings between NN Cr LMs. The Cr clusters develop with increasing Cr concentration; most Cr atoms belong to a big cluster and only a small number of Cr atoms are isolated at 70 at% Fe. The Cr LMs of the cluster are disordered in directions due to the frustration caused by the competition between the AF Cr–Cr NN couplings and the AF Cr–Fe NN couplings, although the Fe LMs still show the ferromagnetic alignment. The developed Cr clusters form a network consisting of the NN Cr bonds around 60 at% Fe, which spreads throughout the MD unit cell as shown in Fig. 8.32. There the Cr LMs of the network are highly disordered in directions. The Fe LMs, on the other hand, are aligned ferromagnetically, though they are slightly fluctuated in directions by the coupling to the disordered Cr LMs. The disorder of Cr LMs in both direction and amplitude continues until 50 at% Fe as shown in Fig. 8.33. The distribution of LMs becomes wider with increasing Cr concentration due to more competition between the F and AF magnetic interactions. At 50 at% Fe, the partial distribution of the Cr LMs is spread from −1.5μB to 1.5μB , and the distribution of the Fe LMs (around 2.0μB ) is also spread, though less broad as compared with the Cr LMs’ at this concentration. These distributions originate mainly in the local-environment effects (LEE), since it is observed from Fig. 8.33 that Cr LMs surrounded by a few Cr

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Fig. 8.32 Magnetic structure of Fe60 Cr40 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms

Fig. 8.33 Magnetic structure of Fe50 Cr50 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms

LMs have large amplitudes, while those surrounded by a few Fe LMs tend to have small amplitudes. The broad distribution of the ferromagnetic Fe LMs continues until 20 at% Fe, where the ferromagnetic order of Fe atoms terminates. At 40 at% Fe, the Cr magnetic moments start to develop the long-range AF order, although some LMs are still fluctuated in directions (broken AF). The ferromagnetic long-range order (LRO) of the Fe coexists with the broken AF of the Cr there. Below 40 at% Fe, the large ferromagnetic Fe NN network which spreads throughout the unit cell starts to split into finite clusters. We find that Fe atoms still form the ferromagnetic LRO at 30 at% Fe. At this concentration, the distribution of Cr LMs

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247

Fig. 8.34 Magnetic structure of Fe20 Cr80 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms. The Fe clusters are shown by the nearest-neighbor (NN) Fe networks drawn by lines and dots on the Fe atoms

are separated into those of up-spin and down-spin sublattices due to the long-range AF order. At 20 at% Fe, the coexistence of the collinear F of Fe and the AF of Cr are observed as shown in Fig. 8.34, where the Cr LMs of up-spin and down-spin sublattices are distributed rather symmetrically. All the magnetizations of separated Fe clusters are oriented in the same direction, due to the long-range ferromagnetic interactions between the Fe LMs. Therefore the effective interaction between the Fe clusters is also ferromagnetic at this concentration. The coexistence of the two long-range orders around 20 at% Fe is consistent with the phase diagram proposed by Loegel [202] and Rode et al. [203]. The AF LRO of the Cr is developed with further increase in Cr concentration. At 15 at% Fe, about one third of the Fe atoms are isolated while the others form ferromagnetic clusters. However, the effective ferromagnetic interactions between the Fe clusters seem weak, because some magnetizations of Fe clusters are opposite to those of the other clusters, as shown in Fig. 8.35. It should be noted that the MD simulations lead to different magnetizations in direction for some Fe clusters when we start with different initial configurations at this concentration. This means that the cluster spin glass (SG) state is possible due to the existence of many degenerate quasi-stable states around this concentration. Moreover, the cluster SG in this concentration region is collinear due to the existence of AF LRO in the present calculations. With increasing further the Cr concentration, the ferromagnetic Fe clusters shrink and most Fe atoms become isolated. At 10 at% Fe, the isolated Fe LMs are mostly aligned not to violate the AF LRO of the Cr matrix, though the Fe LMs are modulated in direction due to long-range Fe–Fe magnetic interactions, as shown in Fig. 8.36. In the range between 5 and 20 at% Fe, every isolated NN Fe pair in the AF Cr sublattices stays in a quasi-stable state in which Fe LMs with different amplitudes

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Fig. 8.35 Magnetic structure of Fe15 Cr85 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms. The Fe clusters are shown by the NN Fe networks drawn by lines and dots on the Fe atoms

Fig. 8.36 Magnetic structure of Fe10 Cr90 alloy obtained by the MD calculations at 25 K [201]. The bright (dark) spheres represent the Cr (Fe) atoms. The Fe clusters are shown by the NN Fe networks drawn by lines and dots on the Fe atoms

are ferromagnetically coupled. One of the LMs of the Fe pair points in the same direction with that of the corresponding sublattice LM and has a larger amplitude, while another LM points in the opposite direction and has a smaller amplitude, as shown in Figs. 8.35 and 8.36. These isolated NN Fe pairs may correspond to the free spin-pairs with equal amplitudes proposed by Friedel and Hedman [209] to explain the Mössbauer line spectra of free rotating iron moments reported at low temperatures. Each isolated Fe pair state with non-equal amplitudes must have a degenerate state whose LMs are exchanged and flipped simultaneously. The potential barrier between these degenerate states in the antiferromagnetic matrix must be low, because no potential barrier arises when we rotate a pair of isolated NN Fe LMs with the same amplitudes, and the energy difference between the isolated NN pairs with

8.4 Computer Simulations for Disordered Magnetic Alloys

249

Fig. 8.37 Concentration dependence of various average magnetic moments in Fe–Cr alloys calculated by the MD approach at 25 K [201]. The dashed line with  represents the average magnetization [m]c , and the dash-dotted (dotted) line with  (•) represents the average Fe (Cr) magnetic moments [mzFe ]c ([mzCr ]c ). Experimental data of these quantities are shown by , , and  with error bars [157, 158, 210–212]

equal LMs and those with non-equal LMs must be small. The NN Fe pairs with non-equal amplitudes are also found in the Fe clusters consisting of more than three atoms (see Figs. 8.34, 8.35, and 8.36). Average magnetic moments of Fe–Cr alloys calculated by the MD approach are presented in Fig. 8.37. The calculated magnetization [m]c decreases gradually as the Fe concentration x is decreased and vanishes at x ≈ 15 at% Fe in good agreement with the experimental data and the Hartree–Fock CPA results (see Fig. 8.5) in the wide range of concentration (30  x ≤ 100 at% Fe). Average amplitudes of Fe and Cr LMs in the pure Cr limit are 1.0μB and 0.7μB , respectively, while in the pure Fe limit, they are 2.35μB and 1.9μB . As the Fe concentration is reduced, the average moment of Fe atoms [mFe ]c increases slightly until 2.4μB at 90 at% Fe, and decreases very slowly to 2.3μB at 60 at% Fe. Below 60 at% Fe, the Fe LMs are subject to the strong LEE, and [mFe ]c changes to slightly smaller values in the range 20  x  60 at% Fe. The calculated average moment of Cr atoms [mCr ]c , on the other hand, starts to gradually increase from the value −1.8μB in the pure Fe limit and vanishes at x ≈ 40 at% Fe. [mCr ]c agrees semi-quantitatively with the experimental results. It should be noted that the vanishing of [mCr ]c in the range 0  x  40 at% Fe is accompanied by the formation of antiferromagnetic or broken antiferromagnetic states, as mentioned before. The average magnitude of the sublattice moments of Cr atoms, in our MD calculation, is small, but does not vanish in the concentration region around x = 20 at% Fe. In the range 25  x  40 at% Fe, Cr LMs continuously change their configurations from the antiferromagnetically ordered state to less ordered states with increasing Fe concentration. The phase diagram of Fe–Cr alloys obtained from the MD calculations at low temperatures is summarized in Fig. 8.38. The alloys are characterized by a persis-

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Magnetism of Disordered Alloys

Fig. 8.38 The magnetic phase diagram of Fe–Cr alloys as a function of Fe concentration obtained from the MD calculations at 25 K [201]. The upper row shows the diagram based on the simple category: the Ferromagnetism (F) and the antiferromagnetism (AF). AF + F denotes the coexistence of the AF and the F. The middle and lower rows are the diagram showing various magnetic structures of each component

Fig. 8.39 The magnetic structure of Fe20 Cr65 Mn15 alloys obtained by the MD calculations at 25 K. The blue, green, and red spheres represent the Fe, Cr, and Mn atoms, respectively. The arrows show their LM’s in arbitrary unit

tence of the magnetic LRO of the constituent atoms up to high concentrations; they show the ferromagnetic LRO of Fe atoms up to 80 at% Cr, and the antiferromagnetic LRO of Cr atoms up to 30 at% Fe. In the narrow region around 20 at% Fe, the F LRO due to Fe LMs coexists with the AF LRO of Cr LMs. The result is consistent with the experimental phase diagram proposed by Loegel [202] and Rode et al. [203]. The wide range of the F state is divided into three regimes, i.e., (i) F with isolated antiparallel Cr LMs (80 at% Fe  x), (ii) F with Cr clusters whose LMs are non-collinearly disordered (50  x  75 at% Fe), and (iii) F with broad distributions of Fe LMs and with non-collinear or broken antiferromagnetic Cr LMs (25  x  45 at% Fe). Furthermore, in the Cr rich regions, the cluster SG states ex-

8.4 Computer Simulations for Disordered Magnetic Alloys

251

ist around 15 at% Fe in the antiferromagnetic LRO of Cr. The SG behaviors around 15 at% Fe are verified experimentally [204, 205]. The MD calculations of ternary alloys are also straightforward. There we construct the random atomic configuration for a given concentration and atomic short range order, and solve the iso-thermal MD equations (6.63). Taking the time average in the equilibrium state, we obtain the average magnetic moment at each site. The magnetic structure of bcc Fe20 Cr80−x Mnx (0 ≤ x ≤ 40) ternary alloys for example has been calculated at 25 K with use of 432 atoms in the MD unit cell (6 × 6 × 6 bcc lattice) [213]. As we have mentioned, Fe20 Cr80 alloys show the coexistence of the Cr AF LRO and the Fe ferromagnetic clusters (see Fig. 8.34 and AF + F in Fig. 8.38). When Cr or Fe atoms are replaced by Mn atoms, the Mn LMs are arranged to enhance the Cr AF LRO, so that the AF structure is stabilized with increasing Mn concentration (see Fig. 8.39). The couplings between the Fe clusters and the Cr–Mn system with the AF LRO are rather weak, so that they form the cluster SG. The MD calculations show that the SG order of Fe ferromagnetic clusters is collinear due to the weak AF coupling between Fe and Cr LMs, and thus the isotropic SG is suppressed in this system.

Chapter 9

Magnetism of Amorphous Metals and Alloys

Amorphous metals and alloys are characterized by the structural disorder which breaks the Bravais lattice. Their magnetic properties are well defined once the quenching method and rate are specified, though their structures are metastable thermodynamically. The amorphous metals and alloys provide us with a fundamental problem of the structure vs. magnetism in the condensed matter physics. In this chapter, we present the theoretical aspects on the magnetism of amorphous metals and alloys. We first give an introduction to the amorphous metallic magnetism in Sect. 9.1. Next we explain how to make the structure model of amorphous metals and alloys, and describe the method to calculate their electronic structure in Sect. 9.2. In Sect. 9.3, we present the finite-temperature theory of magnetism for amorphous metals, and clarify in Sect. 9.4 the basic properties of amorphous transition metals. These properties include the spin glass in amorphous Fe, the enhancement of the Curie temperature of amorphous Co, and the weak ferromagnetism of amorphous Ni. We extend the theory to amorphous alloys in Sect. 9.5, and elucidate in Sect. 9.6 the magnetism of amorphous transition metal alloys as well as the rare-earth transition metal amorphous alloys.

9.1 Introduction to the Amorphous Metallic Magnetism Amorphous alloys are obtained by rapid quenching techniques such as vapor quenching and liquid quenching typically with the cooling rate 106 K/s. Their X-ray diffraction patterns show a halo, indicating the microscopic structural disorder which violates the Bravais lattice [214, 215]. Historically amorphous transition metal alloys containing considerable amounts of metalloids (typically 20 at.% B or P) were investigated at early stage [216–218]. These alloys show a uniform reduction of the magnetization and Curie temperature (TC ) as compared with those of the crystalline alloys. Amorphous Fe80 B10 P10 alloy [218], for example, shows the ferromagnetism with the ground-state magnetization M = 2.1μB and TC = 640 K, while the bcc Fe has M = 2.2μB and TC = 1040 K. Amorphous Co80 B10 P10 alY. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6_9, © Springer-Verlag Berlin Heidelberg 2012

253

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Magnetism of Amorphous Metals and Alloys

Fig. 9.1 Generalized Slater–Pauling plot of magnetization per atom for amorphous Co1−x Bx alloys. Experimental data are shown by open circles [219] and closed circles [220]. The solid line shows the curve obtained from (9.1) with 2Nsp↑ = 0.85

loy also shows the ferromagnetism with M = 1.1μB and TC = 770 K, which are compared with M = 1.7μB and TC = 1388 K in the fcc Co. Magnetization vs. concentration curves were analyzed by using a generalized Slater–Pauling curve picture, which was explained in Sect. 8.1. There we expressed the magnetization M by means of the up-spin valence electron number N↑ and (8.1)). These quantities the total valence electron number Z as M = 2N↑ − Z (see  α +N may be expressed by those of constituent atoms as N↑ = α cα Nd↑ sp↑ and  α Z = α cα Zα . Here cα , Nd↑ , and Nsp↑ are the concentration of atom α, the d electron number with up spin of atom α, and the averaged sp electrons with up spin, respectively. Zα denotes the chemical valence for constituent atom α. Substituting the expressions of N↑ and Z into M = 2N↑ − Z, we obtain M = Zm + 2Nsp↑ .

(9.1)

 α − Z ). Here Zm is the averaged magnetic valence defined by Zm = α cα (2Nd↑ α Since Zm has a simple concentration dependence for the transition metal alloys with strong ferromagnetism, we can expect a linear relation between M and Zm . Figure 9.1 shows the experimental data vs. theoretical curves for amorphous Co1−x Bx alloys. We find a linear relation with the slope one between M and Zm as expected from (9.1). Many amorphous transition metal (TM) metalloid alloys follows a simple generalized Slater–Pauling curve given by (9.1). On the other hand, amorphous transition metal alloys containing early transition metals (ETM) or rare-earth (RE) metals show a dramatic change in their magnetism. In Fe-rich amorphous alloys, the ferromagnetism collapses with increasing Fe concentration, and the spin glass (SG) appears beyond 90 at% Fe as shown in Fig. 9.2 [221]. It is remarkable that the SG appear irrespective of the second el-

9.2 Amorphous Structure and Electronic Structure

255

Fig. 9.2 Magnetic phase diagram showing the Curie temperatures (solid curves) and spin-glass temperatures (dotted curves) of amorphous Fec M1−c alloys (M = La, Zr, Ce, Lu, and Y) [221]

ements, suggesting the SG in amorphous pure iron, which is quite different from the ferromagnetism in bcc Fe. In Co-rich Co–Y amorphous alloys, the Curie temperatures TC are enhanced as compared with those in crystalline counterparts, and rapidly increase with decreasing Y concentration as shown in Fig. 9.3. Extrapolated value of TC to amorphous pure Co reaches 1850 K, which is 450 K higher than that of the fcc Co [222]. These drastic changes of magnetism in the vicinity of amorphous pure metals have revealed the significance of the structural disorder in the amorphous metallic magnetism. One needs microscopic theories of amorphous metallic magnetism to understand these anomalies. In the following section, we will briefly clarify the characteristics of amorphous structure and explain the theoretical approach to calculate the electronic structure of amorphous metals and alloys.

9.2 Amorphous Structure and Electronic Structure In crystalline systems, we obtain the crystal structure and lattice constants from the Bragg peaks in X-ray diffraction, and thus we can calculate the electronic structure with use of the Bloch theorem. This procedure is no longer applicable to amorphous metals and alloys due to the lack of the data on amorphous structure. The X-ray experimental techniques give us only the pair distribution function (PDF), gαγ (R), which is defined by gαγ (R) =

ραγ (R) . ργ

(9.2)

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Fig. 9.3 Curie temperatures in crystalline (solid circles) and amorphous (open circles) Coc Y1−c alloys [222]. The dotted line is an extrapolation to amorphous pure Co

Here ραγ (R) is the density of γ atom at the distance R from an α atom and ργ is the density of γ atom. The distribution function gαγ (R) converges to one as R goes to infinity. Although we can obtain further information on the amorphous structure from EXAFS (extended X-ray absorption fine structure) and neutron measurements, it is not possible to determine experimentally all the atomic positions in the amorphous structure, which are indispensable for electronic-structure calculations of amorphous systems and microscopic understanding of magnetic properties. Thus we have to construct a reasonable model for amorphous structure. Theoretical determination of the amorphous structure is based on the thermodynamical molecular-dynamics (MD) method [107]. In this method, the constituent atoms are distributed in a box with a periodic boundary condition, and the Newton equations of atomic motion are solved by assuming appropriate short-range interatomic pair potentials. A rapid quenching is simulated by reducing the kinetic energy, which is proportional to the temperature, every constant time length, under the condition that either pressure or volume is constant. In the ab-initio MD method [223], we calculate the interatomic forces directly from electronic and atomic structures without any empirical parameters. The ab-initio MD calculations are accelerated by simultaneously solving the equations of motions for both atoms and electrons. The method is called the Car–Parrinello method, and it is most efficient when combined with the pseudo-potential technique and the plain wave orbitals [224]. A more conventional method to construct the amorphous structure for metals is the static method. There we construct the dense random packing of hard spheres (DRPHS) model using the computer, and calculate the PDF. When the calculated PDF does not agree with the experimental one, we relax successively the structure through the atomic force produced by appropriate pair potentials until the agreement is achieved. This is called the relaxed DRPHS model.

9.2 Amorphous Structure and Electronic Structure

257

Fig. 9.4 Pair-distribution function of computer-generated amorphous iron [225]

Figure 9.4 shows an example of a pair distribution function for amorphous Fe obtained by the relaxed DRPHS method [225]. We see that the first peak at r1 = 2.54 Å is sharp, and well separated from the second and third ones. This means that there exists a well-defined nearest-neighbor (NN) shell even in amorphous systems. The ratio of the fluctuation of the NN interatomic distance to the average NN distance is estimated from the width of the first peak to be 0.067, which is in agreement with the experimental data for Fe-rich amorphous alloys. The second peak at r2 = 1.67r1 is considered to originate in the local structures of the rhombi which consist of two regular triangles with a side r1 and the hexahedra which consist of two tetrahedra with the same sides r1 . The third peak at r3 = 2r1 is associated with three contact atoms on a line. Although the amorphous metals and alloys do not form the Bravais lattice, it is possible to construct the tight-binding linear muffin-tin orbital (TB-LMTO) Hamiltonian (2.167) for a given amorphous structure within the density functional theory (DFT) because it is derived without assuming translational symmetry. HiLj L = εiL δij δLL + tiLj L (1 − δij δLL ).

(9.3)

Here the atomic level εiL on site i and orbital L and the transfer integral tiLj L between iL and j L are defined by (2.166). The first-principles electronic structure calculations were made by combining the TB-LMTO method with the recursion method because the latter does not require any translational symmetry for electronic structure calculations (see Appendix G). Figure 9.5 shows the calculated density of states (DOS) for amorphous iron [226]. Note that the DOS for amorphous Fe shows the two-peak structure, and the valley near

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Fig. 9.5 Density of states for amorphous iron [226]

the center of the DOS found in bcc Fe disappears (see the DOS in Fig. 2.6), so that the Fermi level is not on the peak of the DOS. This suggests that the ferromagnetism is not necessarily stable in the amorphous Fe.

9.3 Theory of Amorphous Metallic Magnetism The finite-temperature theory of the LEE presented in Sect. 8.3 can be extended to the amorphous metals. In this section, we describe the theory of magnetism in amorphous metals on the basis of the functional integral method [227–230]. The amorphous metals form a structure close to the dense random packing one, and have a well-defined nearest-neighbor (NN) shell, as we discussed in the last section. The magnetic interactions between the local moments (LM’s) in amorphous metals are expected to be rather short-range because the electron scatterings due to structural disorder cause a strong damping of interaction strength as a function of the interatomic distance. Thus we have a physical picture that the central LM is directly influenced by the LMs on the NN shell, but the effect of more distant atoms and their LMs may be treated as an effective medium. In order to construct a theory based on the physical picture mentioned above, we start from the tight-binding Hamiltonian (3.228) and adopt the free energy (8.19) in the static approximation: F = −β

−1

ln

  i



β J˜i dξi e−βE(ξ ) . 4π

(9.4)

9.3 Theory of Amorphous Metallic Magnetism

259

Here we have omitted the transverse spin fluctuations for simplicity, and J˜i = U0 /D + (1 − 1/D)J , D being the orbital degeneracy (D = 5). In the 5-fold equivalent d band model (8.22), the effective potential E(ξ ) is given by (8.23):  

 1  D U˜ i n˜ i (ξ )2 − J˜i ξi2 . (9.5) E(ξ ) = dω f (ω) Im tr ln L−1 − t − π 4 i

In the above expression, the effective Coulomb interaction U˜ i is given by U0 /D + (1 − 1/D)(2U1 − J ). n˜ i (ξ ) is the Hartree–Fock charge on site i by (8.21). The locator matrix L is defined by   −1

1 ˜ 1 ˜ 0  δ δ = z − ε + μ − n ˜ (ξ ) + ξ σ δij δσ σ  . L iσj σ  = L−1 U J i i i i i iσ ij σ σ 2 2

U˜ i = given

(9.6)

The magnetic moment on site i is obtained by taking the derivative of the free energy F with respect to the magnetic field hi on the same site.   dξj ξi e−βE(ξ ) j

mi  = ξi  =  

. −βE(ξ ) dξj e

(9.7)

j

The energy potential E(ξ ) in (9.7) determines the local moment (LM) on site i. In the amorphous magnetic metals, the energy contains two types of disorder. One is the spin disorder which is caused by the thermal spin fluctuations via random exchange potentials {J˜i ξi σ/2} in (9.6). Another is the structural disorder being characteristic of amorphous metallic systems. The latter appears via the atomic potentials {εi0 − μ + U˜ i n˜ i (ξ )/2} and transfer integral integrals {(t)ij = tij }. Consequently, the diagonal disorder in the locator includes both types of disorder, while the off-diagonal disorder in the transfer integral matrix is caused by the structural disorder only. The diagonal disorder is treated in the same way as in the substitutional disordered alloys, by introducing the inverse effective locator Lσ−1 (z) into the first term at the r.h.s. of (9.5). The deviation from the medium is expanded with respect to the sites as has been made in Sect. 8.3 (see (8.71)). The zeroth order in the expansion of the effective potential E(ξ ) is described by the effective medium only. The first-order correction consists of the sum of single-site energy potentials Ei (ξi ), i.e., (8.72):   

 D −1 ln 1 + L−1 Ei (ξi ) = dω f (ω) Im iσ (z, ξ ) − Lσ (z) Fiiσ (z) π σ 1 1 − U˜ i n˜ i (ξi )2 + J˜i ξi2 . 4 4 Here the coherent Green function Fij σ is defined by (8.76): Fij σ =

 −1

−1  L −t . ij σ

(9.8)

(9.9)

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Note that Fij σ is influenced by the structural disorder via the transfer integrals {tij }.  The second-order correction to the effective medium is the pair-interaction terms (i,j ) Φij (ξi , ξj ) (see (8.74)). The pair energy between sites i and j , Φij (ξi , ξj ) is given by (8.75):     D ln 1 − Fij σ Fj iσ t˜iσ (ξi )t˜j σ (ξj ) . (9.10) Φij (ξi , ξj ) = dω f (ω) Im π σ Here t˜iσ (ξi ) is the single-site t matrix defined by (8.77): t˜iσ (ξi ) = −

−1 L−1 iσ − Lσ

−1 1 + (L−1 iσ − Lσ )Fiiσ

(9.11)

.

The t matrix describes the impurity scattering when the impurity potential εi0 − μ + U˜ i n˜ i (ξ )/2 − J˜ξi σ/2 is embedded in the effective medium Lσ−1 . The higher-order terms in the expansion of E(ξ ) are neglected by assuming a small deviation from the effective medium. The energy potential E(ξ ) is then written as follows.   Ei (ξi ) + Φij (ξi , ξj ). (9.12) E(ξ ) = i

(i,j )

Note that the zeroth term has been dropped, since it does not make any contribution to the thermal average (9.7). Next, one neglects the direct pair interactions between the central LM and the LMs outside the NN shell according to the physical picture discussed at the beginning of this section. By making use of the decoupling approximation as well as the molecular-field approximation for the LMs on the NN shell, the magnetic moment (9.7) is written as follows, as have been made in Sect. 8.3 (see (8.85)).  dξ ξ e−βΨ (ξ ) . (9.13) m0  =  dξ e−βΨ (ξ ) The effective potential Ψ (ξ ) for the central LM is given by (8.84): Ψ (ξ ) = E0 (ξ ) +

z  j =1

(a)

Φ0j (ξ ) −

z  j =1

(e)

Φ0j (ξ )

mj  . xj

(9.14)

Here z on the summation denotes the number of atoms on the NN shell. The atomic (a) (e) (ξ ) and Φ0j (ξ ) are defined by (8.82): and exchange pair energies Φ0j  (a)  Φ0j (ξ )

 1 1 Φ0j (ξ, νxj ). = (e) 2 ν=± −ν Φ0j (ξ )

(9.15)

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261

The quantity xj in (9.14) and (9.15) is an amplitude in the single-site approximation, 1/2 which is defined by xj = ξj2 0 , where the average ∼0 is taken with respect to E0 (ξ ). Equations (9.13) and (9.14) show that a flexible central LM ξ is directly influenced by the molecular fields from the LMs {mj } on the NN shell (the third term in (9.14)) and indirectly by the average molecular field from the LMs outside the NN shell via the spin-dependent effective medium Lσ−1 which appears in E0 (ξ ), (a) (e) Φ0j (ξ ), and Φ0j (ξ ). The effective medium Lσ−1 is chosen so that the correction to the single-site energies in E(ξ ) (see (8.73)) becomes as small as possible. This is an extension of the CPA to the structural disorder, and there is a condition that the averaged singlesite t matrix should vanish. 

 t˜iσ (ξ ) s = 0.

(9.16)

Here the single-site t matrix t˜iσ (ξ ) has been given by (9.11).   ([ ]s ) denotes the thermal (structural) average. As we have mentioned in (8.34), the above condition is equivalent to the following CPA equation.  −1 −1 −1  )  s = Fσ . (Liσ − Lσ−1 + F00σ

(9.17)

Here Fσ denotes a structural average of the coherent Green function. The central LM (9.13) depends on the structural disorder outside the NN shell via the coherent Green functions F00σ , F0j σ (= Fj 0σ ), and Fjj σ in (9.14) and (9.17). These Green functions are treated within the Bethe approximation presented in Sect. 8.3. Applying (8.50) and (8.51) on the locator expansion of coherent Green functions with structural disorder, we obtain F00 = L + L



(9.18)

t0j Fj 0 ,

j =0

Fj 0 = L tj 0 F00 + L Sj Fj 0 + L



Tj i Fi0 .

(9.19)

i =j,0

Here we have omitted the spin suffix σ for brevity and neglected the transfer integrals between the central atom and the atoms outside the NN shell. The self-energy Sj (Tj i ) expresses the contribution of the sum of all the paths which start from site j and end at site j (i) without returning to the cluster on the way (see Fig. 9.6). It should be noted that all the information outside the cluster is included in Sj and Tj i . When we take the structural average outside the cluster, we neglect the last term at the r.h.s. of (9.19) (i.e., the Bethe approximation), and replace Sj with S , i.e., the self-energy of the effective medium for structural disorder. We then obtain (see

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Fig. 9.6 Schematic representation of the irreducible path Sj and Tj i

(8.55) and (8.53))  F00σ =

Lσ−1

Fj 0σ =

+

z 

tj20

j =1

Lσ−1 − Sσ

tj 0 Lσ−1

− Sσ

F00σ .

−1 ,

(9.20)

(9.21)

The diagonal Green function Fjj σ on the NN shell, on the other hand, is approximated by the averaged one.  [ρ(ε)]s dε Fσ = [Fjj σ ]s = . (9.22) Lσ−1 − ε Note that the averaged DOS [ρ(ε)]s for noninteracting systems can be calculated by using the ground-state theories described in the last section (see, for example, Fig. 9.5). The effective medium Sσ is determined from the condition that the structural average of the central coherent Green function F00σ should be identical with the neighboring one.  [ρ(ε)]s dε [F00σ ]s = . (9.23) Lσ−1 − ε The central LM in (9.13) is now determined as follows by the number of NN z, the neighboring LM’s {mj } on the NN shell, the transfer integrals yj = tj20 between the central atom and the neighboring atoms, the effective medium Lσ−1 due to the spin fluctuations, and the effective medium Sσ due to the structural disorder outside the NN shell. ! ! !

m0  = m0  z, mj  , tj20 , Lσ−1 , {Sσ } . (9.24) The structural disorder causes the distribution of LM g(mj ) at the neighboring site j , the distribution of the number of NN p(z), and the distribution ps (yj ) for the

9.3 Theory of Amorphous Metallic Magnetism

263

square of the transfer integral. These distributions determine the LM distribution at the central site via relation (9.13). Since the distribution should be identical with those of the surrounding LMs, we obtain the following integral equation for the LM distribution, taking the same steps as in Sect. 8.3 (see (8.87)). g(M) =



 p(z)

z  

 ps (yi ) dyi g(mi ) dmi . δ M − m0 

z

(9.25)

i=1

The effective mediums Lσ−1 and Sσ are self-consistently determined from (9.17) and (9.23): 

 p(z)

z    −1 −1 −1 ps (yi ) dyi g(mi ) dmi = Fσ , Liσ − Lσ−1 + F00σ

z

(9.26)

i=1



 p(z)

F00σ

z

z    ps (yi ) dyi = Fσ .

(9.27)

i=1

The LM distribution g(M), and the effective mediums Lσ−1 and Sσ are selfconsistently determined by solving (9.25), (9.26), and (9.27). This is the finite temperature theory of the local environment effects for amorphous metals. The average magnetization [m]s and the SG order parameter [m2 ]s are obtained from the distribution g(M) as follows.    m s = Mg(M) dM, (9.28)   m2 s =

 M 2 g(M) dM.

(9.29)

Since self-consistent equations (9.25), (9.26), and (9.27) include 2z-fold integrals, it is not easy to solve the equations. We adopt the following decoupling approximation in the self-consistent equations, which is correct up to the second moment:  

n    M 2n+k g(M) dM ≈ m2 s mk s ,

(9.30)



2n+k n  y − [y]s ps (y) dy ≈ (δy)2 s 0k .

(9.31)

Here k = 0 or 1. These are the lowest approximations which take into account fluctuations. [y]s is a mean square of a transfer integral, and [(δy)2 ]s is the fluctuation around [y]s . After making the decoupling approximations (9.30) and (9.31) at the r.h.s. of (9.25), one can substitute the approximate distribution function g(M) into (9.28) and (9.29). We then obtain the self-consistent equations for [m]s and [m2 ]s as

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follows. 

   z  1 [mn ]s [m]s = , p(z) Γ n, z, [m2 ]s 2 [m2n ]s z



(9.32)

n=0

  n  z−n [mn ]s ξ (z, n, k, l) = , Γ (k, n, q)Γ (l, z − n, q) [m2n ]s ξ 2 (z, n, k, l) k=0 l=0  dξ ξ e−βΨ (ξ,z,n,k,l) , ξ (z, n, k, l) =  dξ e−βΨ (ξ,z,n,k,l) (a)

(9.33)

(9.34)

(a)

Ψ (ξ, z, n, k, l) = E(ξ, n) + nΦ+ (ξ, n) + (z − n)Φ− (ξ, n)   [m2 ]1/2 s (e) (e) − (2k − n)Φ+ (ξ, n) + (2l − z + n)Φ− (ξ, n) , x (9.35)   1 [m]s q= . (9.36) 1+ 1/2 2 [m2 ]s Here Γ (n, z, p) is the binomial distribution function defined by [z!/n!(z − n)!] p n (1 − p)z−n . In the present approximation, the local environments inside the NN shell are de1/2 scribed via the NN transfer integrals by the contraction (−[(δR)2 ]s ) of the NN 1/2 interatomic distance R from average value [R]s and the stretch ([(δR)2 ]s ) of distance R. Thus, the local structure for a given z is specified by means of the number of contracted pairs (n) between the central atom and the atoms on the NN shell. Since a local structure is realized with the probability Γ (n, z, 1/2), the averaged LMs are given by (9.32). The magnetic moments in each local structure (n), [mn ]s and [m2n ]s are given by (9.33). The single-site energy and pair energies are also (a) (e) characterized by n, so that the notations E(ξ, n), Φ± (ξ, n), and Φ± (ξ, n) are used in (9.35). Here the subscript +(−) denotes the contracted (stretched) pair. The parameter q defined by (9.36) is interpreted as the probability that the ficti1/2 tious spin [m2 ]s points up on a site of the NN shell. The probability of finding up spins of k among n contracted atoms on the NN shell is then given by Γ (k, n, q), and the probability of finding up spins of l among z − n stretched atoms is given by Γ (l, z − n, q). Therefore, the average LM in the local environment n, [mn ]s is obtained by averaging ξ (z, n, k, l) over the spin configurations (see (9.33)). Here, ξ (z, n, k, l) denotes the central LM in the (z, n, k, l) configuration. Note that the LMs ξ (z, n, k, l) are strongly influenced by their local environments (z, n, k, l) in transition metals and alloys. Such effects are referred as local environment effects (LEE) in amorphous metals.

9.3 Theory of Amorphous Metallic Magnetism

265

In the same way, one obtains from (9.26) a simplified CPA equation,   1 −1 [ξ ]s  −1  2 1/2

Lσ ν ξ  s − Lσ−1 + Fσ−1 1+ν = Fσ , (9.37) 1/2 2 [ξ 2 ]s ν=± and the following equation from (9.27).  1/2   1 −1

−1 −1 [(δθ )2 ]s −1 ∗ L σ − Sσ = Fσ . (9.38) Lσ − z [y]s 1 + ν 2 [θ ]s ν=±  Here z∗ is an effective coordination number, δθ = θ − [θ ]s , and θ = zj yj . Adopting a simple form p(z) = ([z∗ ] + 1 − z∗ )δz,[z∗ ] + (z∗ − [z∗ ])δz,[z∗ ]+1 , [ ] being 1/2 Gauss’s notation, and Heine’s law t (R) ≡ tj 0 ∝ R −κ , we obtain [(δθ )2 ]s /[θ ]s = 1/2 (z∗ − [z∗ ])([z∗ ] + 1 − z∗ )/z∗ + 4κ 2 Δ/z∗ and [(δy)2 ]s /[y]s = 2κΔ. Here Δ = 1/2 [(δR)2 ]s /[R]s . The average coordination number z∗ , the average interatomic dis1/2 tance [R]s , and its fluctuation [(δR)2 ]s are obtained from the theoretical model or the experimental PDF. The input parameters in the present theory are the d electron number, effective exchange energy parameter J˜, the DOS [ρ(ε)]s , average coordination number z∗ , and 1/2 the fluctuation of the interatomic distance [(δR)2 ]s /[R]s . The input DOS [ρ(ε)]s are presented in Fig. 9.7. [m]s , [m2 ]s , Lσ−1 , and Sσ are obtained by solving (9.32), (9.37), and (9.38) self-consistently. The theory describes the ferromagnetism 1/2 1/2 ([m]s = 0 , [m2 ]s = 0), the SG ([m]s = 0, [m2 ]s = 0), and the paramag1/2 2 netism ([m]s = 0, [m ]s = 0). In the theory mentioned above, we assumed the collinear magnetic moments. When we take into account the noncollinear spin arrangements, we start from the magnetic moment (6.60) [231].     4 ξ e−βE(ξ ) dξ j ξj−2 1 + ˜ξ 2 i β J i j   . (9.39) mi  = dξ j ξj−2 e−βE(ξ ) j

Here the energy potential E(ξ ) is given by (6.56). Taking the same steps we obtain the central magnetic moment corresponding to (9.13) as    4 −2 1+ ξ e−βΨ (ξ ) dξ ξ β J˜ξ 2  m0  = , (9.40) dξ ξ −2 e−βΨ (ξ ) Ψ (ξ ) = E0 (ξ ) + +

 (α,γ )

z   j =1 (b)

(a)

Φ0j (ξ ) −

Φ0j δ (ξ )

 α

(e)

Φ0j α (ξ )

mj α  a˜ j α

mj α mj γ  mj x mjy mj z  (c) . + Φ0j (ξ ) a˜ j α a˜ j γ a˜ j x a˜ jy a˜ j z

(9.41)

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Magnetism of Amorphous Metals and Alloys (a)

Here E0 (ξ ) is the single-site energy, and Φ0j (ξ ) is the atomic type of energy. There (e)

(b)

(c)

are three types of exchange energies, Φ0j α (ξ ), Φ0j δ (ξ ), and Φ0j (ξ ). The amplitude 1/2 a˜ j α is defined by a˜ j α = (1 + 4/β J˜ξj2 0 )ξj2α 0 , ∼0 being the average taken with respect to the single-site energy E0 (ξ ). Equation (9.40) indicates that the central LM is determined by the coordination number z on the NN shell, neighboring LM’s {mj }, square of transfer integrals {yj = tj20 }, effective medium Lσ−1 due to spin fluctuations, and the effective medium Sσ due to structural disorder outside the NN shell: m0  = m0 (z, {mj }, {yj }, {Sσ }, {Lσ−1 }). Thus, the self-consistent equation to determine the distribution function (9.25) is extended as g(M) =



 p(z)

z  

 ps (yj ) dyj g(mj ) dmj . δ M − m0 

(9.42)

j =1

z

Here g(m) is the distribution function for the vector magnetic moments {mi }. The average magnetization [mz ]s and the SG order parameters for each direc1/2 tion [mα 2 ]s (α = x, y, z) are obtained from the distribution g(M) as follows:    mz  s = Mz g(M) dM, (9.43)   mα 2 s =

 Mα2 g(M) dM.

(9.44)

By making use of the decoupling approximation such as (9.30) and (9.31) at the r.h.s. of (9.42) and substituting the approximate expression of g(M) into (9.43) and (9.44), we obtain the self-consistent equations to determine the magnetization 1/2 1/2 1/2 [mz ]s , and the SG order parameters [mz 2 ]s and [mx 2 ]s (= [my 2 ]s ). Since the transverse components of local moments are taken into account, the the1/2 ory describes the noncollinear ferromagnetism ([mz ]s = 0, [mα 2 ]s = 0 (α = 1/2 1/2 x, y, z)), the collinear ferromagnetism ([mz ]s = 0, [mz 2 ]s = 0, [mα 2 ]s = 0 1/2 2 (α = x, y)), the noncollinear SG ([mz ]s = 0, [mα  ]s = 0 (α = x, y, z)), the 1/2 1/2 collinear SG ([mz ]s = 0, [mz 2 ]s = 0, [mα 2 ]s = 0 (α = x, y)), and the para1/2 magnetism ([mz ]s = 0, [mα 2 ]s = 0 (α = x, y, z)). In numerical calculations we apply the Monte-Carlo sampling method because there are too many configurations in the self-consistent equations [231].

9.4 Magnetism of Amorphous Transition Metals Amorphous pure transition metals have not yet been realized in experiments. Nevertheless theoretical results of their magnetism are indispensable for understanding the effects of structural disorder on the metallic magnetism as well as the experimental data of amorphous alloys. In this section, we elucidate the theoretical results

9.4 Magnetism of Amorphous Transition Metals

267

Fig. 9.7 Input DOS per atom for the amorphous (solid curve), bcc (dot-dashed curve), and fcc (dotted curve) structures

obtained by the finite-temperature theory of amorphous metals and discuss on the consistency of the results with experimental data [227, 228, 230, 231].

9.4.1 General Survey The crystalline Fe, Co, Ni transition metals are well-known to show a simple ferromagnetism with uniform magnetization. There the Stoner condition for the stability of the ferromagnetism provides us with a useful criterion as discussed in Sect. 2.3.4: ρ(0)J˜/2 > 1, where ρ(0) denotes the noninteracting density of states (DOS) per atom at the Fermi level. Note that we adopted here the DOS per atom instead of the DOS per atom per spin for convenience. Figure 9.7 shows the DOS for the amorphous, bcc, and fcc structures. The ferromagnetism of bcc Fe, for example, is stabilized by the main peak near the Fermi level. The fcc Ni shows the strong ferromagnetism for the same reason. A characteristic feature of the DOS for amorphous transition metals is that the main peak is located just between the bcc and fcc peaks. The d electron numbers N ∗ with the Fermi level at each main peak are N ∗ (bcc) = 7.44 for the bcc, N ∗ (amor) = 8.35 for amorphous structure, and N ∗ (fcc) = 9.05 for the fcc. They indicate the strong ferromagnetic region in each structure. The Stoner criterion for amorphous Fe is estimated to be ρ(0)J˜/2 = 0.96 < 1 with use of the effective exchange energy parameter J˜ = 0.068 Ry (see Table 2.1) and the first-principles DOS

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Magnetism of Amorphous Metals and Alloys

Fig. 9.8 Calculated magnetization vs. d-electron number curves at 75 K for the amorphous (solid curve), bcc (dot-dashed curves), and fcc (dotted curve) structures [228]. The effective exchange energy parameter is fixed to that of the bcc Fe. The dashed curve shows the spin-glass (SG) order parameter. The d-electron numbers N ∗ integrated up to the main peak of each DOS in Fig. 9.7 are indicated by the arrows. The inset shows the experimental data for amorphous (Fe–M)90 Zr10 alloys. Here, M = Mn (), Co (), and Ni (◦) [232]

for amorphous Fe [226]. The result suggests that amorphous Fe does not develop a uniform magnetization. In the same way, the ferromagnetism in amorphous Co is expected to be enhanced because the main peak is near the Fermi level, while amorphous Ni may not maintain strong ferromagnetism because its Fermi level is located above the peak position of the amorphous DOS. Another characteristic of the DOS for amorphous structure is their similarity to the fcc DOS rather than the bcc ones, particularly near the top of the d bands, because the amorphous structure is close to the close-packed structure. The amorphous transition metals and alloys are therefore expected to show anomalous magnetic properties similar to those found in the fcc ones. For example, we have seen in Sect. 8.3 that the Fe-base fcc alloys have nonlinear magnetic couplings: the ferromagnetic couplings when the amplitudes of Fe LM’s are large and the antiferromagnetic couplings when the amplitudes are small. This peculiarity may explain the SG states in Fe-rich amorphous alloys. The Invar anomalies as found in Fe-base amorphous alloys are also expected because of the similarity in electronic structure. The Stoner model is based on the assumption of the uniform magnetization which is not generally satisfied in the amorphous metallic system. To examine the existence of the ferromagnetism, one needs to perform numerical calculations using the theory presented in the last section. Figure 9.8 shows the calculated magnetization vs. delectron number curves. There the input parameters are set as those of amorphous iron except for the d electron number (N ) to see the qualitative feature. We find that the N ∗ , which are indicated by arrows, correspond well to the magnetic regions of

9.4 Magnetism of Amorphous Transition Metals

269

Fig. 9.9 Curie temperature vs. d-electron number curves calculated with use of the same input parameters as in Fig. 9.8 [228]. The SG temperatures are shown by the dashed curve. The d-electron numbers N ∗ are indicated by arrows

the d electron number for each structure. It verifies that the 3d ferromagnetism is stabilized by the magnetic energy gain associated with the main peak in the DOS for each structure. The Curie temperature maxima are also characterized by N ∗ , as shown in Fig. 9.9. According to thermodynamic considerations, the Curie temperature TC is roughly given by TC ∼ Emag /Smag . Here Emag is the magnetic energy defined by the energy difference between the ground state and the paramagnetic state, and Smag is the magnetic entropy. The result in Fig. 9.9 indicates that the Curie temperatures for transition metals showing strong ferromagnetism are dominated by the magnetic energy gain associated with the main peak for each structure. This explains qualitatively why the Curie temperature of amorphous Co is enhanced. It is found that the paramagnetic susceptibility at high temperatures for each structure also shows the enhancement near the electron number N ∗ (bcc) = 7.44, N ∗ (amor) = 8.35, and N ∗ (fcc) = 9.05, respectively. It implies that the susceptibility enhancement due to the magnetic energy gain associated with the DOS at the Fermi level remains even above TC , in spite of large thermal spin fluctuations. The experimental data for liquid transition metal alloys support such a physical picture [233, 234]. One of the important features of the magnetism in amorphous transition metals is that the SG solution ([m2 ]s = 0 and [m]s = 0) appears, as shown in Figs. 9.8 and 9.9. Although the SG solutions exist even for Co (N ≈ 8.0), they are not realized because the ferromagnetic state is more stable. The SG state is expected to be stabilized in the region N  7.35, where the ferromagnetism disappears. In the region N  6.7 we find antiferromagnetic NN interactions irrespective of the local environment, so that the antiferromagnetism should be stabilized there, though their magnetic structure is not well known.

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Magnetism of Amorphous Metals and Alloys

Fig. 9.10 Magnetic phase diagram as a function of the d-electron number N around amorphous Fe showing the ferromagnetic (F), paramagnetic (P), and spin-glass (SG) states [228]. The cluster SG (CSG) phase is expected to fall in the region 7.2 < N < 7.385

9.4.2 Magnetism of Amorphous Fe, Co, and Ni As expected from the general survey of amorphous transition metals, the amorphous Fe shows the SG. Figure 9.10 shows the magnetic phase diagram obtained by the finite-temperature theory (collinear case) presented in Sect. 9.3. The SG states with the transition temperatures Tg = 100–200 K are extended to a reasonable range of amorphous Fe (6.7  N  7.35). The transition temperature Tg = 117 K is obtained 1/2 for N = 7.0 when the experimental value [(δR)2 ]s /[R]s = 0.067 is applied. It is consistent with the observed Tg = 120 K. The formation of SG’s is understood from the behavior of the exchange pair (e) energy −Φ0j (ξ ) in Ψ (ξ ) in the nonpolarized medium (see (9.14)). The energy (e)

−Φj 0 (ξ ) is interpreted as the magnetic pair-energy gain for the flexible central local moment (LM) ξ when the neighboring LM with average amplitude xj points up, as (e) seen from (9.35). Figure 9.11 shows the exchange pair energies −Φ0j (ξ ) in various environments in the SG state. In contrast to the bcc Fe (see Fig. 8.22(b)), neighboring Fe LMs show nonlinear magnetic couplings for the environments 3  n  12; the Fe LMs with large amplitudes couple ferromagnetically with their neighbors, while the Fe LMs with small amplitudes couple antiferromagnetically with the neighboring ones. This behavior has also been found in fcc Fe crystalline alloys (see Fig. 8.22(a)). Since the amplitudes of LM depend strongly on the surrounding environments via the single-site energy E(ξ, n) as shown in Fig. 9.11, the sign of the magnetic couplings changes with the local environments. The picture obtained for Fe LMs in various environments is shown in Fig. 9.12. The competition between these ferro- and antiferro-magnetic couplings produces the itinerant-electron

9.4 Magnetism of Amorphous Transition Metals

271

(e) Fig. 9.11 Single-site energy E(ξ, n) (dotted curves) and exchange pair energy −Φ+ (ξ, n) (solid curves) of amorphous Fe in various environments (n) at T = 35 K [228]. Here n denotes the number of contracted pairs

Fig. 9.12 Schematic representation showing the local environment effect on the central local moment (LM) in amorphous Fe. n denotes the number of contracted atoms on the NN shell

SG in amorphous iron. The mechanism mentioned above is characteristic of amorphous metallic magnetism, since neither the local environment effects (LEE) on the amplitude of LM nor the non-linearity of the magnetic couplings are seen in the insulators. Numerical calculations show that anomalous nonlinear magnetic couplings appear in the region 6.7  N  7.2. When the d electron number N is increased further, the NN couplings do not show any non-linearity irrespective of the local environments at 7.2  N . Nevertheless, the SG state is found in the region 7.2  N  7.385. This is explained as follows. The LM’s form local ferromagnetic orders according to the NN ferromagnetic couplings. If these ferromagnetic orders developed a long-range ferromagnetic order, the effective medium would have a polarization consistent with the ferromagnetic clusters. However, the antiferromagnetic couplings between the central LM and the medium occur when the medium is polarized. These couplings reverse the central LM’s, so that a long-range ferromagnetic order cannot be developed. This implies that the competition between the short-range ferromagnetic couplings and the long-range antiferromagnetic couplings produces the SG in the region 7.2  N  7.35. Since the SG is accompanied

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Magnetism of Amorphous Metals and Alloys

Fig. 9.13 The LM distributions g(M) as a function of the d-electron number at 35 K

by the ferromagnetic clusters, it is referred to as a cluster SG. As the d-electron number is increased, the size of the clusters is expected to increase more and more. Finally, a long-range ferromagnetic order is realized in the region 7.35  N . Such a physical picture is consistent with the neutron experiments on reentrant Fe90 Zr10 amorphous alloys showing a coexistence of propagating spin-wave excitations and spin-freezing phenomena [235]. Calculated LMs show broad distributions in the vicinity of the ferromagnetic instability and the SG region, as shown in Fig. 9.13. They are caused by the LEE on both the amplitudes and directions of LMs. The results are consistent with the broad distributions of the hyperfine field found in amorphous Fe93 Zr7 [238], Fe92 La8 [239], and Fe92 Hf8 [240] alloys. The most intriguing feature in the calculated magnetic phase diagram of Fig. 9.10 is the reentrant SG behavior in the narrow region 7.350  N  7.385. The magnetic states in this region are determined by a detailed balance between the shortrange ferromagnetic interactions and the long-range antiferromagnetic interactions. Theoretical calculations showed that the reentrant SG behavior is caused by the temperature-induced enhancement of the short-range ferromagnetic couplings with increasing temperature [228]. The reentrant SG produced by the structural disorder is considered to be realized in Fe-rich amorphous alloys around 90 at.% Fe (see Fig. 9.2). The noncollinear theory of amorphous metallic magnetism given in the second half of the last section yields similar results [231]. Figure 9.14 shows the calculated magnetic moments as a function of d electron number N . The magnetization [mz ]s shows a maximum at a d electron number around N = 7.6 and rapidly decreases with decreasing d electron number towards amorphous Fe (N ∼ 7.0). The ferromagnetism becomes noncollinear in the region 7.38 ≤ N ≤ 7.43 as shown in 1/2 Fig. 9.15. There the transverse SG order parameter [mx 2 ]s becomes finite in the

9.4 Magnetism of Amorphous Transition Metals

273

1/2

Fig. 9.14 Total spin glass order parameter [m2 ]s , the transverse spin glass order parameter 1/2 [mx 2 ]s , and the magnetization [mz ]s as functions of d electron number N at 30 K [231] Fig. 9.15 Distribution of local moments at 30 K for d electron numbers N = 7.00 (left) and N = 7.42 (right) [231]. Here 4000 data points among 32000 Monte Carlo samplings are shown

presence of the magnetization [mz ]s . At N = 7.38, the magnetization disappears 1/2 1/2 while both total and transverse SG order parameters ([m2 ]s and [mx 2 ]s ) remain finite, showing the second-order transition from the noncollinear ferromagnetism (F) to the noncollinear SG. The SG region is divided into two regimes as in the collinear theory. In the region 7.2  N ≤ 7.38, the cluster SG accompanied by the ferromagnetic clusters is realized, while the SG due to the nonlinear magnetic coupling between NN LMs are realized in the region 6.9  N  7.2. The SG in amorphous Fe (N ∼ 7.0) is caused by this mechanism. The LM distribution is nearly spherical as shown in Fig. 9.15. It should be noted that the distribution of LMs deviates from the spherical one with decreasing N and shows nearly two-dimensional disc shaped distribution in the vicinity of N = 6.9 where the SG order parameter shows a minimum. The magnetic phase diagram obtained by the noncollinear theory is presented in Fig. 9.16. The Curie temperature TC rapidly decreases with decreasing the d electron number and reaches the multicritical point at N = 7.38 and T = 104 K. Note that

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9

Magnetism of Amorphous Metals and Alloys

Fig. 9.16 Magnetic phase diagram as a function of temperature T and d electron number N , showing the paramagnetic (P), the collinear ferromagnetic (coll. F), the noncollinear ferromagnetic (noncoll. F), and the spin glass (SG) states [231]

both TC and Tg in the noncollinear theory are reduced approximately by a factor of two as compared with the collinear ones because of the semi-classical treatment of spins in the theory. In the noncollinear theory, the re-entrant F–SG transition is not found. Instead in the narrow region around N = 7.42, there is a transverse 1/2 spin freezing temperature Tf at which the transverse SG order parameter [mx 2 ]s appears. The SG transition temperature Tg shows a minimum as a function of N around N = 6.9, where the average NN magnetic interactions change the sign. The noncollinear SG is stable in the region 6.9  N ≤ 7.38 while the collinear and the noncollinear SG are almost degenerate for N  6.9. Experimental phase diagrams seem to be consistent with that of the collinear theory (i.e., Fig. 9.10), suggesting the existence of local anisotropy in the real systems. The first-principles ground-state calculations for amorphous Co have been performed on the basis of the tight-binding LMTO-recursion method [241]. It is verified that the Fermi level of nonmagnetic amorphous Co is just at the main peak as shown in Fig. 9.17, so that strong ferromagnetism is realized, as discussed before. The calculated ground-state magnetizations are 1.63μB for amorphous Co, 1.58μB for fcc Co, and 1.55μB for hcp Co. The experimental values are reported to be 1.72μB for both amorphous and hcp Co. Subtracting the orbital contribution 0.15μB from the experimental data, we find the spin contribution 1.57μB , being in good agreement with the theoretical results. As mentioned before, the Curie temperature of amorphous Co obtained by an extrapolation of the data for amorphous Co–Y alloys amounts to 1850 K, which is 450 K higher than that of crystalline Co (see Fig. 9.3). The spin wave stiffness constant is also enhanced by 330 [meV·Å2 ] as compared with that of the hcp Co (510 [meV·Å2 ]), indicating the enhancement of the magnetic coupling due to the

9.4 Magnetism of Amorphous Transition Metals

275

Fig. 9.17 The up and down DOS for amorphous Co at the ground state. The dotted curve shows the DOS per spin in the nonmagnetic state [241]

Fig. 9.18 Calculated magnetizations, inverse susceptibilities, and amplitudes of LM’s for amorphous (solid curves) and fcc (dotted curves) Co (N = 8.1 and J˜ = 0.100 Ry) as a function of temperature [228]

structural disorder [222]. The finite-temperature theory of amorphous metallic magnetism allows us to understand the anomaly qualitatively or semiquantitatively as shown in Fig. 9.18. The calculated TC are overestimated by a factor of 1.8 due to the single-site approximation. Theoretical result of TC for amorphous Co is 490 K

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Fig. 9.19 Ferro- (F) and para-magnetic (P) phase boundaries for the amorphous (solid curve), bcc (dot-dashed curve), and fcc (dotted curve) structures on the J˜–N plane, which are obtained by making use of the Stoner theory [229]

higher than that of fcc Co. This is because the magnetic energy gain due to the main peak at the Fermi level enhances TC in amorphous Co, while the ground-state magnetization is approximately the same for fcc Co because of strong ferromagnetism in both amorphous and fcc Co. The same type of enhancement of TC is verified to occur in the range 7.9  N  8.5 in accordance with N ∗ (amor) = 8.35. The strong ferromagnetism in Ni is expected to be weakened with the introduction of structural disorder, since the main peak near the top of the d band is broadened and shifts down to the lower energy region so that the Fermi level is located above the peak. Figure 9.19 shows a diagram for the Stoner ferromagnetism near Ni as a function of the d electron number N , which is obtained with use of the DOS in Fig. 9.5 after a scaling of the band width by a factor of W (Fe)/W (Ni) = 0.393/0.364. A weak ferromagnetism or paramagnetism is obtained around N = 9.0 and the LDA value J˜ = 0.074 Ry (see Table 2.1). Experimentally, the ground state magnetization, 0.45μB , and TC = 480 K are obtained by an extrapolation of the data for amorphous Nix Y1−x (0.75  x  0.97) [236, 237]. The data, however, do not converge to the same value beyond 90 at.% Ni when Y is replaced by La [239]. On the other hand, the negative value of the experimental Weiss constant obtained from the inverse susceptibility in liquid Ni suggests that Ni with the liquid structure would be paramagnetic at zero temperature [233, 234]. These scattered data indicate the importance of the degree of structural disorder which will be discussed in the next subsection.

9.4.3 Degree of Structural Disorder and Nonunique Magnetism It is obvious that different microscopic random structures are possible for the same amorphous metal. Therefore the magnetic properties may change depending on the

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277

degree of structural disorder. Experimentally, different structures can be created by the size difference between the transition metal atom and the second elements, the difference in chemical bond with the second elements, and the different preparation techniques. In fact, the magnetization of amorphous Fe obtained by an extrapolation of the magnetization vs. concentration curve is 2.2μB in amorphous Fec B1−c alloys [242, 243], though the Fe-rich amorphous alloys containing early transition metals (ET) show the SG beyond 90 at% Fe (see Fig. 9.2). Amorphous Fe powders containing 2 wt% H, 3 wt% C, and 1 wt% O show the ferromagnetism having the magnetic moment about 1.4μB at the ground state [244]. The same controversial results are also found in amorphous Ni. The Curie temperatures TC in Ni–Y [236, 237], Ni–La [239], sputtered Ni–Zr [245], and meltquenched Ni–Zr amorphous alloys [246] do not agree with each other even beyond 90 at% Ni, and yield different values when they are extrapolated to amorphous pure Ni. From the analysis of Ni–B amorphous alloys, the pure amorphous Ni is expected to be ferromagnetic with a saturation magnetization of about 60 % of that of the crystalline counterpart and to have the TC lower by about 60 K [247]. The finite temperature theory of amorphous metallic magnetism presented in the last section allows us to investigate the nonunique magnetism mentioned above as a function of the average coordination number z∗ and the fluctuation of the inter1/2 atomic distance Δ1/2 = [(δR)2 ]s /[R]s . This is understood from the fact that we can obtain the magnetic moments [m]s and [m2 ]s by solving (9.32) and (9.37) self-consistently, once we know the coordination number z∗ , the fluctuation of interatomic distance Δ1/2 , the average band width z∗ [y]s , and the effective self-energy outside the cluster {Sσ }. The latter two quantities are obtained from the average densities of states (DOS) [ρ(ε)]s for the noninteracting system via (9.38) and the following relations (see (9.22)).  [ρ(ε)]s dε Fσ = , (9.45) Lσ−1 − ε    (9.46) z∗ [y]s = ε 2 ρ(ε) s dε. A remarkable point of the theory is that it describes the magnetism in both amorphous and crystalline structures. In order to describe the magnetism in the intermediate regime of the degree of structural disorder, we adopt a simple interpolation scheme for z∗ [y]s and Sσ .

(9.47) z∗ [y]s = A + B z∗ − za∗ + CΔ, ∗

Sσ = Aσ + Bσ z − za∗ + Cσ Δ. (9.48) The coefficients are determined from the values at three points on the z∗ − Δ plane: the crystalline bcc (zb∗ = 8, Δb = 0), the fcc (zf∗ = 12, Δf = 0), and an amorphous 1/2 structure (za∗ = 11.5, Δa = 0.067), for which the DOS [ρ(ε)]s are known. These points b (= bcc), f (= fcc), and a (= amor) are shown in Fig. 9.20 by closed circles.

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Fig. 9.20 Magnetic phase diagram of Fe on the z∗ − Δ plane at 50 K [230]. F, SG, and P indicate the ferromagnetism, the spin glass, and the paramagnetism, respectively. The bcc, fcc, sputtered amorphous Fe, and powder amorphous Fe are shown by closed circles and open circles. Note that calculations are not made in the right-triangle region at the upper left corner, since the region is far from the reference points specified by the closed circles

Figure 9.20 shows the calculated magnetic phase diagram of Fe on the z∗ − Δ plane at 50 K [230]. Three magnetic phases are possible in the present calculations: the ferromagnetic (F), the SG, and the paramagnetic (P) states depending on the degree of structural disorder. The phase boundary between F and SG may be characterized by a line z∗ ≈ 10.5, because the average ferromagnetic couplings are mainly governed by the coordination number z∗ . A paramagnetic region appears around z∗ ≈ 11.0 and Δ ≈ 0, because the average magnetic coupling vanishes and fluctuations due to structural disorder are small there. The complex magnetic structures such as spin-density waves are expected near the fcc Fe (z∗ = 12, Δ = 0) due to long-range competing magnetic interactions, though the present theory does not describe them since the long-range magnetic couplings are not taken into account. The magnetic phase diagram explains some of the non-unique experimental data of magnetic moment and TC of amorphous Fe. The Fe-rich early-transition metal (ET) amorphous alloys with more than 90 at.% Fe are reported to have za∗ ≈ 11.5 and 1/2 Δa ≈ 0.067 [225, 248]. These values lead to the SG of amorphous Fe according to the phase diagram, Fig. 9.20, in agreement with the experimental result. We obtain Tg = 125 K at this point, which is comparable to the experimental value 110 K (see Fig. 9.2). Figure 9.21 shows various magnetic moments of amorphous Fe and static spin correlations between the NN LMs at 50 K as a function of Δ along the straight line between the bcc point and the amor point (see Fig. 9.20). Calculated magnetization gradually decreases first with introducing structural disorder. It begins to show the ferromagnetic instability beyond Δ1/2 ≈ 0.05, and finally disappears at Δ1/2 = 0.058. At this point, Fe shows the transition from the ferromagnetism to the 1/2 SG, since the SG order parameter [m2 ]s remains beyond Δ1/2 = 0.058 as seen

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279

Fig. 9.21 Various magnetic moments of Fe ([m]s : the 1/2 lower solid curve, [m2 ]s : 1/2 2 dotted curve, [m ]s : the upper solid curve) at 50 K, and the effective Bohr magneton number (meff : dotted curve) along the bcc-amor line in Fig. 9.20 [230]. The nearest-neighbor (NN) static spin correlation at 1/2 50 K ([m0 m1 ]s /[m2 ]s ) is also presented by the thin solid curve

in Fig. 9.21. These behaviors are explained by the gradual decrease of noninteracting DOS at the Fermi level caused by the shift of the main peak to the higher energy region. Note that the ferromagnetic instability occurs before the Stoner instability point (Δ1/2 = 0.0635) where the condition [ρ(0)]s J˜/2 = 1 is satisfied. This indicates that the disappearance of the ferromagnetism is realized by the reversal of LMs with increasing Δ. The ferromagnetism with the ground state magnetization M = 1.4μB and TC /TC (bcc) 0.557 is reported for amorphous Fe powder containing H, C, and O impurities [244]. Experimental data of the radial distribution function yield the values z∗ ≈ 10.5 and Δ1/2 ≈ 0.075 [244]. These values lead to the ferromagnetism according to the phase diagram (Fig. 9.20). We obtain there M = 2.02μB and TC /TC (bcc) = 0.437, which are consistent with the experimental data mentioned above. The amorphous Fe expected from Fe–metalloid alloys seems to show the ferromagnetism. Since the metalloid atoms are much smaller than Fe atoms, they tend to occupy the interstitial position between Fe atoms, so that the packing fraction may be smaller than that expected from Fe–ET amorphous alloys. This means that amorphous Fe obtained from the Fe–metalloid alloys are rather close to the bcc structure in coordination number z∗ . The theoretical phase diagram and these structural considerations explain the ferromagnetism of the ‘pure’ amorphous Fe obtained by an extrapolation from the amorphous Fe–metalloid alloys, though the volume effect and the change of electronic structure due to the metalloid are also significant in this system. In the case of Ni, the phase diagram consists of the paramagnetic (P) and ferromagnetic (F) states as shown in Fig. 9.22 [249]. The phase boundary is located around z∗ ≈ 11.0, though the results are rather sensitive to the d electron number N . Since the average coordination numbers z∗ in typical amorphous Ni are expected to be around 11.0, the magnetism of amorphous Ni is sensitive to the degree of structural disorder as well as the other parameters. Detailed numerical calculations verify that both the ground-state magnetization and TC in disordered Ni are smaller than those of fcc Ni. The result is consistent with experimental facts.

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Fig. 9.22 Magnetic phase diagram of Ni on the z∗ − Δ plane at 50 K for the d electron number N = 9.0 (solid curve) and 9.1 (dot-dashed curve) [249]. Note that the calculations are not made in the right-triangle region at the upper left

The ground-state magnetization [m]s and TC of amorphous pure Ni obtained from the Ni–B amorphous alloys are [m]s ≈ 0.37μB and TC ≈ 580 K [247]. With use of the estimated structural parameters, z∗ ≈ 11.75 and Δ1/2 ≈ 0.060 for amorphous Ni, we obtain the theoretical result [m]s = 0.40μB and TC = 380 K, being consistent with the experimental data. The Curie temperatures of the melt-quenched amorphous Ni91 Zr9 alloy [246] and its counterpart in the bcc phase are practically identical and are much smaller than those obtained by the sputtering method [245]. We can explain the behavior from the phase diagram as a difference in coordination numbers between the two: z∗ (melt-quenched) < z∗ (sputtered), though one has to take into account the other factors such as the effects of second elements, atomic volume and the chemical short-range order. The concentration dependence of the Weiss constants in liquid Fe–Ni alloys suggests that Ni with liquid structure is nonmagnetic at the ground state. As is well-known, the structure of liquid Ni is characterized by z∗ ≈ 10.5 and Δ1/2 ≈ 0.083. The phase diagram (Fig. 9.2) explains the paramagnetism of liquid Ni if Δ1/2 ≈ 0.083 and z∗  10.5.

9.5 Theory of Magnetism in Amorphous Alloys It is remarkable that all the metallic glasses are realized in amorphous alloys where both structural and configurational disorders play an important role on their magnetic properties. In this section, we extend the finite-temperature theory of amorphous metallic magnetism to the amorphous alloys in order to understand the magnetism of amorphous transition metal (TM) alloys. The structure of amorphous alloys is characterized by structural disorder and random atomic configuration of the constituent atoms. To specify the local structure of

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281

these amorphous alloys, we need more detailed microscopic parameters. The average coordination number on the nearest-neighbor (NN) shell depends on the type of the central atom α as zα∗ , due to the difference in atomic size of the constituent ∗ < z∗ when the atomic size of atom A is atoms. We expect that for example zA B smaller than that of B. Accordingly the atomic short-range order (ASRO) parameters also depend on the type of atom α as τα , so that the probability of finding atom α at the neighboring site of atom α is given by p αα = cα + (1 − cα )τα .

(9.49)

The consistency for the number of neighboring A–B pairs (i.e., cA zB∗ p AB = ∗ p BA ) yields the following relation among {z∗ } and {τ }. cB zA α α ∗ zA (1 − τA ) = zB∗ (1 − τB ).

(9.50)

In the DRPHS-like model we have an approximate but simple relation as follows.   (9.51) zα∗ = zα∗ (0) + p αα zα∗ (1) − zα∗ (0) . Here zα∗ (1) (zα∗ (0)) is the average coordination number for p αα = 1 (p αα = 0). When the relations mentioned above are accepted, the independent parameter which controls the degree of atomic short-range order is either τA or τB . Note that choice of a random configuration τA = τB = 0 is not allowed because of the relation (9.50). The most random configuration is obtained byminimizing the mean square  of deviation taken over all the NN pairs Φ = (N/2) α cα γ zα∗ p αγ (p αγ − cγ )2 under the condition (9.50), so that the following condition is obtained [250]. cA τB + cB τA = 0.

(9.52)

Equations (9.50), (9.51), and (9.52) determine the parameters {τα , zα∗ } in the most random configuration. The Green function theory of electronic structure calculations for substitutional alloys, which was presented in Sect. 8.3, can be extended to the amorphous alloys [250, 251]. We adopt the Hartree–Fock tight-binding Hamiltonian (8.4) and assume the geometrical mean model for the transfer integral model tij = (c)∗ (c) (c) (c)∗ rα tij rγ (see (8.16)). We then consider the Green function Gij = rα Gij rγ = −1

(c) 2 (L−1 − t  ) . Here (L−1 )ij = L−1 j δij = δij (ω + iδ − εi )/|ri | and we have omitted the spin indices for simplicity. By making use of the Dyson equations (8.50) and (8.51) for G , and making the Bethe approximation (see (8.55)), the site-diagonal Green function is obtained as follows. −1  z 2  t 1 ij G00 = (c) . (9.53) L−1 α − L−1 − Sj (L ) |rα |2 j =0 j

Here z is the coordination number on the nearest neighbor shell. Sj (L ) is the sum of all the paths which start from site j and end at the same site without returning to the cluster on the way.

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It should be noted that the transfer integral tij still depends on the atomic configαγ uration on sites i and j via the interatomic distance Rij = Rij in the case of amorαγ γγ phous alloys. Experimentally the geometrical-mean relation Rij = (Rijαα Rij )1/2 holds true within a few percent error [251]. Thus, using the relation and assuming the same power law relation tij ∝ Rij−κ , we find tij = rα(s)∗ tˆij rγ(s) .

(9.54)

Here tˆij is the transfer integral between sites i and j for an amorphous metal as a reference system. When amorphous pure metals A and B have the same structure (s) except for the interatomic distance, the factor rα does not depend on sites i and j . Adopting (9.54), we obtain the Green function G00 as follows.  −1 z  tˆj20 1 −1 G00 = . Lα − (s) |rα |2 L−1 − S  (L)/|r |2 j =0

j

(9.55)

j

j

√ (c) (s) Here rα = rα rα . It is given by rA = μ2 (A)/μ2 (B) and rB = 1, μ2 (α) being moment for the average DOS for amorphous pure metal α: μ2 (α) = the 2second 2 ε [ρα0 (ε)]s dε. Moreover L−1 j = (ω + iδ − εj )/|rj | . When we take the structural and configurational average outside the cluster, we (s) 2 ˆ replace the self-energy Sj (L)/|r j | with an effective S , so that we obtain −1  z  yj 1 −1 . Gαα = Lα − |rα |2 L−1 j −S

(9.56)

j =0

Here yj = tˆj20 . Introducing the probability pα (z) of finding z sites on the NN shell of atom α and the probability ps (yj ) dyj of finding tˆj20 between yj and yj + dyj , we obtain the averaged Green function for the type of atom α as follows.    z z    

αα [Gαα ]s c = pα (z) Γ n, z, p ps (yj ) dyj Gαα . (9.57) z

n=0

j

Here [ ]c ([ ]s ) means the configurational (structural) average. Γ (n, z, p) is the binomial distribution function [z!/n!(z − n)!]p n (1 − p)z−n , and Γ (n, z, p αα ) expresses the probability of finding n atoms of type α on the NN shell of atom α with z sites. By making use of the decoupling approximation (9.31), we obtain z    

[Gαα ]s c = pα (z) Γ n, z, p αα z

×

n=0 n  z−n  i=0 j =0

    1 1 Γ j, z − n, Gαα (z, n, i, j ), Γ i, n, 2 2

(9.58)

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283

  1/2  1 [(δy)2 ]s −1 L − n + (2i − n) [y]s Kα Gαα (z, n, i, j ) = [y]s |rα |2 α −1  1/2  [(δy)2 ]s [y]s Kα¯ − z − n + (2i − z + n) . (9.59) [y]s Here [y]s is obtained from the second moment and the average coordination number 1/2 z∗ as [y]s = μ2 (B)/z∗ . Moreover [(δy)2 ]s /[y]s = 2κΔ, Δ being defined by Δ = 1/2 −1 [(δR)2 ]s /[R]s . The quantity Kα is defined by Kα = (L−1 α −S) . To obtain the effective self-energy S outside the cluster, we consider the coherent Green function in which the locators of the Green function rα∗ G00 rα = [(L−1 − tˆ)−1 ] have been replaced by the effective one L , i.e., F (L −1 ) = [[(L −1 − tˆ)−1 ]00 ]s . In the Bethe approximation it is given as



−1  F L −1 = L −1 − θ K . s

(9.60)

 Here θ = zj yj and K = (L −1 − S )−1 . We thus obtain K and S from the above equation because the l.h.s. is given by (9.22). Kα in (9.59) is obtained from K as

−1 −1 Kα = L−1 − K −1 . α −L

(9.61)

The effective locator L −1 should be determined by the self-consistent equations for the shell boundary condition which corresponds to (8.63) for substitutional alloys.  ∗ 

rj Gjj rj s c = F L −1 . (9.62) Here the Green function rj∗ Gjj rj is given by (8.56): rj∗ Gjj rj =

1 L−1 j

−S

+

1 (L−1 j

− S )2

yj rα∗ G00 rα .

(9.63)

In a more simple version, the medium L −1 is determined by the CPA equation (8.18):  

−1 −1

−1 cα L−1 + F L −1 = F L −1 . (9.64) α −L α

In order to describe the different shapes of the DOS in both pure amorphous metal limits, it is suitable to adopt a common band model such as tˆij = λt˜ij = λ[cA tij (A)+ cB tij (B)]. Here tij (α) is the transfer  integral for the pure amorphous metals α. The parameter λ is determined as λ = μ2 (B)/μ˜ 2 by taking the second moment at both sides of equation tˆij = λt˜ij . μ˜ 2 is the second moment of the average DOS for the common band {t˜ij } and is given by μ˜ 2 = [cA μ2 (A)1/2 + cB μ2 (B)1/2 ]2 . When we adopt the common band model, K in (9.61) is modified as K = (L −1 − λS˜ )−1 ,

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Fig. 9.23 Calculated DOS for amorphous Fe65 Zr35 alloys with the use of the Bethe-type approximation (solid curve) and the tight-binding LMTO recursion method (dotted curve) [251]. The ∗ ) numbers for the solid curves denote the average coordination numbers of Fe atoms (zFe

and S˜ for the transfer integral t˜ij is approximated as S˜ = cA S (A) + cB S (B). Each S (α) or Kα0 = (L −1 − λS (α))−1 for amorphous pure metal α is determined from the following condition corresponding to (9.60).



−1  . Fα L −1 = L −1 − λ2 θα Kα0 s

(9.65)

Thus K in the common band model is obtained from Kα0 as −1 −1 −1 K = cA KA0 + cB KB0 .

(9.66)

Finally, (9.60), (9.64), and (9.66) form the self-consistent equations to obtain L −1 and K at each energy ω + iδ. Once we obtain them, we obtain Kα via (9.61). Thus we can calculate the Green function [[Gαα ]c ]s from (9.58) and (9.59). The densities of states (DOS) of the amorphous Fe65 Zr35 alloy calculated by the above-mentioned Bethe-type theory are shown in Fig. 9.23 [251]. Calculated DOS’s show a two-peak structure near the Fermi level. The low energy peak is higher ∗ = 12. The high-energy peak on the Fermi level than the high-energy peak when zFe ∗ and becomes dominant at z∗ = 8.0. The calrapidly develops with decreasing zFe Fe ∗ = 9.0. The amorculated DOS agrees well with the first-principles DOS when zFe phous Fe65 Zr35 alloy is known to show the ferromagnetism with the ground state magnetization of 0.95μB . Since the high DOS at the Fermi level is important for the appearance of the ferromagnetism according to the Stoner criterion, the results indicate that the ferromagnetism in amorphous Fe65 Zr35 alloy originates in the formation of the high-energy peak due to the atomic-size difference, which is not seen in the substitutional alloys.

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285

The finite-temperature theory of amorphous metallic magnetism is extended to the case of amorphous alloys [250]. In the theory for amorphous alloys, we start from the expression of the magnetic moment on site i given by (9.7). The energy potential E(ξ ) is given by (9.5) in which the locator matrix L has been replaced by −1

L iσj σ  = L−1 iσ δij δσ σ    7 1 1 = ω + iδ − εi0 + μ − U˜ i n˜ i (ξ ) + J˜i ξi σ |rj |2 δij δσ σ  , (9.67) 2 2 and the transfer integral (t)ij with (tˆ)ij . The latter is defined by tij = ri∗ tˆij rj . Taking the same steps as in Sect. 9.3, we introduce the effective medium Lσ−1 and expand the energy E(ξ ) with respect to the site in the medium. After making the decoupling approximation for surrounding field variables as well as the molecularfield type approximation, we reach (9.13) for the central local moment (LM):  dξ ξ e−βΨ (ξ ) . (9.68) m0  = ξ  =  dξ e−βΨ (ξ ) Here Ψ (ξ ) is given by (9.14): Ψ (ξ ) = E0 (ξ ) +

z  j =1

(a)

(a) Φ0j (ξ ) −

z 

(e) Φ0j (ξ )

j =1

mj  . xj

(9.69)

(e)

E0 (ξ ), Φ0j (ξ ), and Φ0j (ξ ) are given by (9.8), (9.10), and (9.15) in which the locators L have been replaced by (9.67) and coherent Green functions Fij σ have been replaced by Fij σ =

 −1

−1  Lσ − tˆ . ij

(9.70)

In the Bethe approximation, they are given by (9.20) and (9.21) in which {tij } have been replaced by {tˆij }. The central LM in (9.68) is regarded as a function of the surrounding LM’s {mj } on the NN shell, the squares of transfer integrals {yj = tˆj20 }, the atomic configuration on the NN shell {γj }, and the coordination number z. These variables randomly change because of the structural and configurational disorders. Thus we introduce a probability gγj (mj ) dmj of finding LM on the atom of type γj between mj and mj + dmj , a probability ps (yj ) dyj of finding tˆj20 between yj and yj + dyj , a probability p αα of finding an atom of type α at the neighboring site of the central atom α, and a probability pα (z) of finding z sites on the NN shell of the central atom α. These determine the distribution function of the central LM mα  via (9.68) and the latter should be identical with those of the neighboring sites. Then the selfconsistent equation for the distribution function is obtained as follows (see (9.25)).

286

gα (M) =

9



Magnetism of Amorphous Metals and Alloys

 z 



αα δ M − mα  pα (z) Γ n, z, p

z

n=0

n z       ps (yi ) dyi gα (mi ) dmi ps (yj ) dyj gα¯ (mj ) dmj . × i=1

(9.71)

i=n+1

By making use of the decoupling approximations (9.30) and (9.31) for the distribution functions gγ (mj ) and ps (yj ) at the r.h.s. of (9.71)  and substituting the approximate expression of gα (M) into [[mα ]s ]c = Mgα (M) dM  and [[mα 2 ]s ]c = M 2 gα (M) dM, we obtain the self-consistent equations for [[mα ]s ]c and [[mα 2 ]s ]c as follows.     z n 

 1 [[mα ]s ]c αα = p (z) Γ n, z, p Γ i, n, α [[mα 2 ]s ]c 2 z n=0

×

z−n  j =0



Γ j, z − n,

1 2

i=0

[[ξα (z, n, i, j )]s ]c . [[ξα 2 (z, n, i, j )]s ]c



(9.72)

In the approximate expression, the central LM of atom, i.e., [[ξα k (z, n, i, j )]s ]c (k = 1, 2) at the r.h.s. of (9.72) are specified by the coordination number (z), the number of α atoms on the NN shell (n), the number of contracted atoms among n atoms of type α on the shell (i), and the number of contracted atoms among the remaining z − n atoms of type α(j ¯ ). The distribution function pα (z) is given in the simplest form as follows.

   

(9.73) pα (z) = zα∗ + 1 − zα∗ δz[zα∗ ] + zα∗ − zα∗ δz[zα∗ ]+1 , where [ ] denotes Gauss’ notation. The LMs [[ξα k (z, n, i, j )]s ]c (k = 1, 2) are obtained by taking the average over 1/2 the configurations of fictitious spins {[[mα 2 ]s ]c } on the NN shell with use of the probability qα of finding an up spin [250]. Here qα is given by   [[mα ]s ]c 1 qα = . (9.74) 1+ 1/2 2 [[mα 2 ]s ]c The theory covers a wide range of amorphous and liquid magnetic alloys, from metals to insulators within a molecular-field approximation. In the limit of the pure amorphous metal, it reduces to the theory presented in Sect. 9.3. The theory reduces to the finite-temperature theory of the LEE presented in Sect. 8.3 in the case of substitutional alloys. There numerical investigations revealed the following points: (1) the magnetizations at low temperatures are semi-quantitatively described by the effective exchange energy parameters J˜α ; (2) the concentration dependences of Curie temperatures are reproduced, but the absolute values are overestimated by a factor of 1.5–2.0 due to the molecular-field approximation and the neglect of transverse spin fluctuations; and (3) the effective Bohr magneton numbers obtained from the Curie–Weiss susceptibilities are generally underestimated by about 25 %

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287

because of the overestimated itinerant character due to the use of reduced J˜α and static approximation. The theory takes into account the fluctuations of the LM’s with respect to the structural and configurational disorders. This leads to the spin-glass (SG) solution ([[mα ]s ]c = 0 and [[mα 2 ]s ]c = 0), in addition to the ferromagnetic ([[mα ]s ]c = 0 and [[mα 2 ]s ]c = 0) and paramagnetic ([[mα ]s ]c = [[mα 2 ]s ]c = 0) solutions. In the local-moment limit, the SG transition temperature reduces to (8.100) for the case of substitutional alloys:

 !  1 2 2 2 2 2 4 1/2 . + cB JBB + cA JAA − cB JBB + 4cA cB JAB Tg2 = z cA JAA 2 (9.75) For amorphous metals, it is given by

1 (9.76) Tg2 = z∗ J+2 + J−2 . 2 Here z∗ is the average coordination number, Jαγ is the exchange coupling energy between α and γ atoms, and J+ (J− ) is the exchange coupling for a contracted (stretched) pair. Equation (9.75) is in agreement with the SG temperature Tg for an insulator model in the molecular field approximation (see (7.46)). Equation (9.76) √ reduces to Tg for the ±J model, i.e., Tg = z∗ |J | when J = J+ = −J− . These facts show that the theory describes both types of SG on the same footing, i.e., the SG due to configurational disorder and the SG due to structural disorder. Needless to say, the latter is essential for amorphous magnetism.

9.6 Magnetism of Amorphous Transition Metal Alloys Amorphous transition metal (TM) alloys show a variety of magnetic properties due to structural and configurational disorders. In this section, we elucidate their properties on the basis of their electronic structure and the finite-temperature theory of magnetism for amorphous alloys.

9.6.1 TM–TM Amorphous Alloys Amorphous TM–TM alloys are significant from the theoretical point of view because they enable us to clarify the effects of structural disorder by comparing their magnetic properties with those of substitutional alloys. We first discuss the magnetism of Fe–Ni amorphous alloys. Experimentally these alloys are found in rather narrow concentration regime 0.64 ≤ cFe ≤ 0.72 [252]. In the crystalline Fe–Ni alloys, the saturation magnetization abruptly decreases around 65 at% Fe (see Sect. 8.3). There the Invar anomalies such as the zero thermal expansion and a broad hyperfine field distribution occurs. In contrast to the crystalline case, the hyperfine field in amorphous Fe–Ni films was found to show no anomaly around

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Fig. 9.24 Concentration dependence of the magnetic moment at 150 K and the Curie temperature calculated by a single-site theory for amorphous (solid lines), fcc (dashed lines), and bcc (dot-dashed lines) Fe–Ni alloys [254]. The inset is the corresponding hyperfine field observed experimentally [252]

the Invar concentration range in contrast to the crystalline case. Instead it increases along the Slater–Pauling curve with increasing Fe concentration, as shown in the inset of Fig. 9.24. Magnetic properties of amorphous Fe–Ni alloys have been calculated by means of the single-site theory of amorphous magnetic alloys with use of the geometrical mean model [254]. The results are shown in Fig. 9.24. A remarkable point is that the critical concentration of the ferromagnetic instability shifts to the Fe side by about 30 at% Fe, so that the strong ferromagnetism is realized around 70 at% Fe. It originates from the change of electronic structure. In fact, the shift of the main peak to the lower energy region in the nonmagnetic DOS, which was found in amorphous pure metals (see Fig. 9.7), remains even in amorphous alloys, therefore amorphous Fe–Ni alloys have the Fermi level at the main peak around 70 at% Fe, whereas their crystalline fcc counterparts have the Fermi level below the peak in the same concentration region. The change of the main peak due to structural disorder leads to the strong ferromagnetism in the Invar concentrations in amorphous Fe–Ni alloys. The shift of the Curie temperature maximum to the higher Fe concentration due to structural disorder is also explained by the magnetic energy gain associated with the shift of the main peak in the DOS. Most of the TM–TM magnetic alloys are formed by adding 4d and 5d early transition metals (ET = Y, Zr, Nb, Mo, La, Hf, Ta, and W) to 3d magnetic transition metals (TM = Fe, Co, and Ni). These alloys are characterized by a large difference in atomic size between the ET and the 3d TM, so that the NN atomic distance depends strongly on the type of the NN pair and that the coordination number depends

9.6 Magnetism of Amorphous Transition Metal Alloys

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Fig. 9.25 Concentration dependencies of ASRO parameters {τα } (solid lines) and the average coordination numbers {zα∗ } (dashed lines) in amorphous Fe–Zr alloys, which are obtained from the condition of the most random atomic configuration (see (9.52)) and a linear equation for {zα∗ } (see (9.51)) [250]

strongly on the type of the central atom as well as the atomic short-range order parameters. Figure 9.25 shows the average coordination number {zα∗ } as well as the atomic short range order (ASRO) parameters for amorphous Fe–Zr alloys in the most random atomic configuration (see (9.52)) [250]. Here the coordination numbers in the ∗ (0) = 7.0, z∗ (0) = 16.0, and z∗ (1) = z∗ (1) = 11.5 are estimated pure limits, zFe Zr Fe Zr on the basis of the DRPHS model. The average coordination numbers zα∗ simply decrease with decreasing Fe concentration. An example of calculated DOS for the amorphous Fe75 Zr25 alloy is presented in Fig. 9.26 [250]. The DOS calculated with use of the finite temperature theory at low temperatures agree well with those of the tight-binding LMTO supercell method [253]. The latter leads to the magnetization 1.27μB at T = 0 K and the former to the magnetization 0.9μB at 75 K. These values are compared with the experimental data 0.96μB . Note that Zr atoms are also polarized to be anti-parallel to the magnetization because of the strong hybridization of Zr 4d bands with 3d Fe minority-spin bands. The appearance of the ferromagnetism is explained by a large difference in atomic size between Fe and Zr atoms, as discussed in the last section (see Fig. 9.23). The atomic short range order is also important for the magnetism of amorphous Fe–Zr alloys. Figure 9.27 shows the calculated phase diagram on the p FeFe –cFe plane at 75 K. When we decrease the Fe concentration along the line of the most random atomic configuration, we have the spin glass (SG) to the ferromagnetic state (F) and to the paramagnetic state (P) transition. However, we have no ferromagnetic region when p FeFe 0.8, and the SG to P transition is expected with decreasing the Fe concentration in the region p FeFe 0.8. The disappearance of the ferromagnetism is caused by the formation of amorphous Fe clusters accompanied by the 3d band broadening. Experimentally the Curie temperatures TC in the sputtered Fe–Zr

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Fig. 9.26 Calculated average local DOS of amorphous Fe75 Zr25 alloy for Fe (solid curves) and Zr (dashed curves) atoms at 75 K [250]. Dotted curves show the local DOS at zero temperature obtained by the tight-binding LMTO supercell method with 64 atoms in a unit cell [253]

amorphous alloys are larger than those in the melt-span Fe–Zr amorphous alloys. The result is explained by p FeFe (sputtered) < p FeFe (melt-spun) according to the diagram. The calculated magnetic phase diagram on the temperature-concentration plane is shown in Fig. 9.28 [250]. The phase diagram is obtained for the most random atomic configuration. It consists of the three different phases: P, F, and SG. In the narrow range of concentration between the F and the SG, the re-entrant SG (RSG) appears. In the amorphous pure Fe limit, the SG is stable as has been discussed in Sect. 9.4. With decreasing Fe concentration, the ferromagnetism appears because of the atomic-size effects as mentioned in the last section. Calculated TC shows the maximum at 70 at% Fe in agreement with the experimental data (see the inset of Fig. 9.28), and decreases with decreasing Fe concentration. The transition temperatures (TC and Tg ) are overestimated by a factor of 1.5–2.0 as compared with the experimental ones because of the molecular-field approximation. The SG in the Fe-rich region are partly caused by the nonlinear magnetic couplings between the NN Fe LM’s and partly by the competition between short-range ferromagnetic coupling and long-range antiferromagnetic coupling as discussed in the amorphous pure Fe (see Fig. 9.11). In fact in the former case the central Fe LM with more than 5 contracted Fe NN (l > 5) couples antiferromagnetically with (e) the neighboring Fe LMs via −ΦFeFe+ (ξ, l), but the central Fe LM with l < 5 ferromagnetically couples to the neighboring Fe LMs. On the other hand, the correlation between the central Fe LM m0Fe  and the neighboring Fe LM m1Fe , i.e., [[m0Fe m1Fe ]c ]s is 0.15 at 90 at% Fe and 75 K. This means that the NN ferromagnetic interactions are rather strong as compared with the NN antiferromagnetic

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Fig. 9.27 The magnetic phase diagram on the p FeFe –cFe plane at 75 K obtained by the finite-temperature theory [255]. The dot-dashed line denotes the most random atomic configuration (see (9.52)), and the solid curves denote the phase boundaries among the spin glass (SG), the ferromagnetism, and the paramagnetism (P). The dotted line shows the lower bound; p FeFe = 1 − ∗ /c z∗ cZr zZr Fe Fe

Fig. 9.28 Calculated magnetic phase diagram of amorphous Fe–Zr alloys showing the paramagnetism (P), ferromagnetism (F), spin glass (SG), and the re-entrant spin glass (RSG) [255]. The phase boundary below 50 K are extrapolated by dashed lines. The inset shows the experimental result [221, 256]

ones. The SG near the phase boundary therefore should be formed by the competition between short-range ferromagnetic couplings and long-range antiferromagnetic interaction. The Co–Y amorphous alloys show the ferromagnetism and it is enhanced as compared with their crystalline counterparts in a wide range of Co concentrations, as shown in Fig. 9.3. Enhancement of the ferromagnetism at more than about 80 at% Co is explained by the high DOS of the main peak at the Fermi level as found in amorphous pure Co (see Fig. 9.17). However, the enhancement around 50 to 80 at% Co is attributed to the differences in local atomic structure, especially, the small

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Fig. 9.29 Change in DOS for amorphous Co67 Y33 alloy with varying the coordination number ∗ , from 8 to 12 [251]. z∗ is fixed to be 12. The inset shows the result based around a Co atom, zCo Y on the tight-binding d orbital model [258]. The DOS at the Fermi level is developed with decreas∗ ing zCo

atomic coordination number of Co due to the difference in the atomic sizes between Co and Y atoms. We discuss here the Co2 Y alloys as an example. The amorphous Co2 Y alloy shows the ferromagnetism with the average magnetization 0.7μB per atom, while the crystalline Laves-phase Co2 Y shows the paramagnetism (see Fig. 9.3). The first-principles band calculations show that the Fermi level of the Laves-phase Co2 Y is pinned in a local minimum of the DOS, so that the compound does not satisfy the Stoner criterion [257]. On the other hand, the DOS in amorphous Co2 Y alloy show the peak at the Fermi level as shown in the inset of ∗ , Fig. 9.29 [258]. The DOS for various average coordination number of Co, i.e., zCo which are calculated by means of the theory of the LEE of amorphous alloys presented in the last section are shown in Fig. 9.29 [251]. The peak near the Fermi level ∗ . The DOS at the Fermi level increases with decreasing z∗ , strongly depends on zCo Co ∗ = 10 and 9. The coordination number and satisfies the Stoner criterion between zCo around a Co atom is 12 in the Laves phase Co2 Y, whereas it is about 9.5 in its amorphous counterpart. Since the latter satisfies the Stoner criterion, one concludes that the atomic-size difference causes the ferromagnetism in the amorphous Co2 Y alloy. Calculated Curie temperature vs. concentration curves of both amorphous and compound Co–Y alloys are given in Fig. 9.30. The Curie temperatures for the crystalline counterparts are calculated as the close-packed Co–Y alloys having the same average local atomic configurations {zα∗ } and {τα } as those in the crystalline counterparts (i.e., Co2 Y, Co3 Y, Co5 Y, Co17 Y2 , and fcc Co) and no fluctuations of 1/2 Δ = [(δR)2 ]s /[R]s (i.e., Δ = 0). Calculated TC ’s explain qualitatively the difference between the amorphous structure and its Laves phase crystalline counterpart as well as their concentration dependence, though the calculated TC for crystalline Co2 Y alloy is still finite in this approach.

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Fig. 9.30 Calculated Curie temperature (TC ) vs. concentration curve [259]. Closed circles show the Curie temperatures for close-packed alloys with the same average local atomic configurations as in the crystalline fcc Co, Co17 Y2 , Co5 Y, Co3 Y, and Co2 Y compounds. The inset shows the experimental results [222, 260]

The TM–Y alloys form amorphous alloys with all the magnetic TM from Mn to Ni in a wide range of concentration. Their magnetic phase diagrams are obtained experimentally as shown in Fig. 9.31 [261]. Amorphous Mnc Y1−c (0.18 ≤ c ≤ 0.76) alloys show the SG with SG transition temperatures (Tg ) less than 60 K [262, 263]. Amorphous Fec Y1−c (0.32 ≤ c ≤ 0.925) alloys are reported to have two different phase diagrams: the sputtered Fe—Y alloys show the SG in the whole concentration [264–266], while the melt-spun Fe–Y alloys show the ferromagnetism with the reentrant SG behavior in the range 0.5  c  0.8 [267–269]. Amorphous Coc Y1−c (0.38 ≤ c ≤ 0.90) alloys show the enhancement of TC as compared with their crystalline counterparts as has been mentioned. The Nic Y1−c (0.65 ≤ c ≤ 0.97) alloys show a weak ferromagnetism beyond 83 at% Ni [236, 270]. Both alloys cause the F–P (paramagnetism) transition at 50 at% Co and 83 at% Ni, respectively. The overall feature of the magnetic phase diagrams of TM–Y amorphous alloys is explained by the finite temperature theory of the LEE for amorphous alloys as shown in Fig. 9.32 [261]. There the most random atomic configuration is assumed. Calculated TC in amorphous Ni–Y alloys is 500 K for amorphous pure Ni, which is smaller than the TC in fcc Ni (630 K), and monotonically decreases with increasing 3d–4d hybridization. As discussed before, the main peak at the Fermi level in amorphous Co enhances the magnetic coupling, therefore TC . The atomic-size effects tend to keep this enhancement when Co concentration is decreased. The calculated SG temperature in amorphous Mn–Y alloys, on the other hand, linearly increases with increasing Mn concentration, and show a maximum at 80 at% Mn because of the band broadening with increasing Mn concentration. Calculated phase diagram in amorphous Fe–Y alloys shows the SG beyond 87 at% Fe and the re-entrant SG behavior in a very small region near the

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Fig. 9.31 Experimental magnetic phase diagrams in amorphous TM–Y (TM = Mn, Fe, Co, Ni) alloys [222, 262, 263, 267–270]. Solid (dashed) curves denote the Curie (spin-glass) temperature TC (Tg ). The SG temperatures in Mn–Y alloys are multiplied by a factor of 2. The inset shows different phase diagrams for the melt-spun Fe–Y [267–269] and sputtered Fe–Y amorphous alloys [264, 265]

Fig. 9.32 Calculated magnetic phase diagrams in amorphous TM–Y alloys [230]. The solid line in Fe–Y beyond 87 at% Fe and the dotted line show the SG temperature. The other solid curves show the Curie temperature

F–SG boundary. Calculated Tg changes from 240 K (amorphous Fe) to 165 K (86.5 at% Fe). Below 86 at% Fe, the alloys show the ferromagnetism. The Curie temperature rapidly increases with decreasing Fe concentration, and show a maximum (TC = 740 K) around 60 at% Fe. These results qualitatively agree with the

9.6 Magnetism of Amorphous Transition Metal Alloys

295

Fig. 9.33 The DOS for amorphous Fe80 B20 alloys (solid curve), and for crystalline Fe2 B compound (dashed curve) [272]

magnetic phase diagram in melt-spun Fe–Y alloys (see Fig. 9.31), though the reentrant SG behaviors are not found in a wide range of concentration. The sputtered amorphous Fe–Y alloys do not show the ferromagnetism in whole concentrations according to the experiments. It is explained by the ASRO effects. In fact, theoretical calculations verify that the amorphous Fe–Y alloys show the SG irrespective of concentration when the condition p FeFe 0.8 is satisfied. Since the experimental values of p FeFe seem to satisfy the condition (for example, p FeFe = 0.845 for the sputtered Fe85 Y15 alloys), the SG in sputtered Fe–Y alloys are explained by the formation of small Fe clusters due to strong ASRO.

9.6.2 The Other TM Alloys The transition metal metalloid amorphous alloys have also been intensely investigated from the earliest stage of investigations [216–218]. Their local structures are shown to be similar to those of the crystalline counterparts. For example, there is no direct contact between metalloid P atoms in Co–P amorphous alloys from 0 at% P to about 20 at% P, and metalloid atoms are captured in the interstitial positions called the Bernal holes in the DRPHS model [271]. The local structure is quite similar to that of trigonal-prismatic crystalline Fe3 P and Fe2 P compounds. Figure 9.33 shows the calculated DOS for the amorphous Fe80 B20 alloy and the crystalline Fe2 B [272]. The characteristic feature of these DOS is that they are almost the same. Their electronic structure based on the first-principles calculations is summarized as follows. The B sp states hybridize strongly with Fe d states. The hybridization splits B sp states into bonding and antibonding states, and shifts the main peak of DOS for Fe d states toward the low energy side. The Fermi level

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Fig. 9.34 The average magnetization per TM atom for the pseudo-binary transition metal amorphous alloys as a function of valence electron number [273]. The crystalline Slater–Pauling curves are also plotted by dashed lines

becomes close to the main peak position of the DOS. The Fe–Fe distances increase, and the coordination numbers of Fe–Fe pairs decrease with the addition of B atoms, so that the d band width of Fe narrows. These effects are common for TM-metalloid alloys. In particular, the peak shift of the DOS toward the low energy side enhances the DOS of Fe at the Fermi level, so that the ferromagnetism is stabilized according to the Stoner criterion in the Fe–metalloid amorphous alloys. The effects of the peak shift of the DOS are also found in the other TM-metalloid amorphous alloys. Figure 9.34 shows the average magnetization vs. d electron number curves for pseudo-binary transition metal metalloid alloys in the form (TM1x TM21−x )80 B10 P10 [273]. It is remarkable that the curves for amorphous alloys are obtained simply by shifting the corresponding curves for crystalline alloys by 0.5 electrons to lower d-electron numbers. The shift of the electron number corresponds to that of the peak position of the DOS with the addition of metalloids. We have discussed the same kind of systematic change of the magnetism in amorphous TM metals in Sect. 9.4. The origin of the shift of the DOS here, however, is different from the case of amorphous pure metals; the shift of the DOS in the TM-metalloid system is caused by the change in chemical bondings between TM and metalloid atoms, while the shift in pure TM amorphous metals is caused by structural disorder. The rare-earth transition metal (RE–TM) amorphous alloys have also extensively been investigated. Their atomic structures are similar to those obtained by the DRPHS model, which are different from the crystalline counterparts. The valence electron configurations and atomic radii of RE atoms are quite similar to those of a Y atom. The itinerant-electron contributions to the magnetic properties are similar to those of Y-TM amorphous alloys. The RE atoms on the other hand have unfilled f-electron shell inside 6s2 filled shell, so that they have well-defined atomic LMs even in solids [16]. The magnetic moment of an RE atom is given by M = L + 2S (μB ), while the total angular momentum J is given by J = L + S. Here L and S

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297

denote the total atomic orbital and spin angular momenta of f electrons, respectively. These atomic moments are built up by the Hund’s rule coupling, i.e., the Coulomb interactions between f electrons, except for Ce showing the valence fluctuations. In the case of RE atoms, the spin–orbit coupling λL · S is significant, and the coupling constant λ takes plus (minus) sign for the f electron number less (more) than half in filling (see (1.44) in Sect. 1.3). Therefore the atoms form the multiplet ground state with J = L − S for the RE atoms with less than half-filled f electrons (i.e., light RE atoms) and J = L + S for the RE atoms with more than half-filled f electrons (i.e, heavy RE atoms). Between the 3d TM atomic spin S TM and the RE atomic spin S, there is an effective interatomic interaction of the type −JRT S · S TM . To derive the interaction, assume that the 3d bands are ferromagnetically polarized. Since the broad 5d up-spin band electrons strongly hybridize with the 3d down-spin band electrons as compared with the hybridization with 3d up-spin bands, the 5d electrons are polarized antiferromagnetically. The latter electrons polarize f electrons in the same direction due to the direct exchange interaction on RE sites. This implies that the effective coupling between TM spin S TM and RE spin S is antiferromagnetic; JRT < 0. The crystalline-field effects on the RE LM are usually assumed to be described by the single-ion anisotropy such as −Di (ni · J i )2 [214, 215]. Here the random unit vector ni denotes a local ‘easy axis’ on site i and Di (> 0) is the anisotropy constant. Since the local easy axis is random, this interaction is favorable for the noncollinear magnetic structure. Because of the magnetic interactions mentioned above, the RE-TM amorphous alloys are classified into the following three groups. Light RE–TM amorphous alloys The local moment (LM) of TM (TM = Fe, Co, and Ni) atoms couples antiferromagnetically to the spin S of the RE atoms and the latter antiferromagnetically couples with the orbital LM L (λ > 0). Since the orbital LM exceeds the spin LM in magnitude, it turns out that the total LM of RE atoms ferromagnetically couple to those of the TM atoms. The random single-ion anisotropic interactions cant the RE LM to some extent. Therefore this system shows non-collinear ferromagnetism as shown in Fig. 9.35(left). The Nd–Fe and Nd–Co amorphous alloys belong to this category. Heavy RE–TM amorphous alloys The LM of TM (TM = Fe, Co, and Ni) atoms couples antiferromagnetically to the spin S of the RE atoms and the latter ferromagnetically couples with the orbital LM L (λ < 0). Thus the total LM of the RE atoms antiferromagnetically couple to those of the TM atoms. The random single-ion anisotropic interactions cant the RE LM to some extent. Therefore this system shows non-collinear ferrimagnetism as shown in Fig. 9.35(right). Amorphous Dy–Fe and Dy–Co alloys for example belong to this category. Gd–TM amorphous alloys The LM of TM (TM = Fe, Co, and Ni) atoms couples antiferromagnetically to the spin S of Gd atoms. Since the orbital LM L for Gd atoms disappear, the total LM (i.e, spin S) of the RE atoms antiferromagnetically couple to those of the TM

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Fig. 9.35 Three types of magnetic configurations in amorphous RE–TM alloys: light RE–TM alloys (left), Gd–TM alloys (middle), and heavy RE–TM alloys (right). The solid (dashed) arrows denote the TM (RE) LM

atoms. This system therefore shows the ferrimagnetism as shown in the middle of Fig. 9.35. Amorphous Gd–Fe, Gd–Co, and Gd–Ni alloys show the ferrimagnetism. The first-principles calculations have been performed for some RE–TM amorphous alloys with the use of the density functional theory (DFT) with the local density approximation (LDA). Figure 9.36 shows the densities of states (DOS) for the amorphous Gd33 Fe67 alloy [274]. The Fe 3d states ferromagnetically polarize as in the case of bcc Fe and the projected DOS show the two-peak structure. The twopeak structure originates in the local atomic structure which is rather more similar to the closed packed structure than the crystalline bcc or the Laves structure. The Gd 5d states extend from −3 eV to 8 eV and strongly hybridize with the minority spin 3d band because the latter is energetically closer to the 5d states than the majority 3d states, as seen in Fig. 9.36. Consequently, the 5d electrons polarize to be antiparallel to the 3d LM’s. The localized Gd 4f states form the narrow up and down bands. Since the polarized 5d electrons ferromagnetically couple to the 4f electrons via the intraatomic d–f exchange interaction, the 4f electrons antiparallely couple to the 3d Fe LM’s as mentioned before. Note that the DFT-LDA theory underestimates the

Fig. 9.36 Projected DOS’s of amorphous Gd33 Fe67 alloy [274]

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binding energy of the occupied f states; the calculated value 4.3 eV is much smaller than the experimental value 9.4 eV [275]. The calculated magnetic moments in the amorphous Gd33 Fe67 alloy are 7.2μB for the Gd site and −2.0μB for the Fe site. The average magnetic moment per atom is 1.1μB , which is in good agreement with the experimental value 1.2μB . In the case of the crystalline Laves phase GdFe2 compound, calculated magnetic moments of the Gd site are 7.5μB , which is larger than in the amorphous alloy, and −1.9μB for Fe site. The average magnetic moment per atom is 1.2μB , which is also in good agreement with the experimental value (0.9–1.2μB ) [276]. In order to discuss the temperature dependence of magnetic properties of the RE–TM amorphous alloys, one has to rely on the model Hamiltonian. A possible effective Hamiltonian which takes into account the itinerant character of d electrons as well as the localized f electrons may be written as follows [277–279].     † εi niσ + tij aiσ aj σ + Ui ni↑ ni↓ − J˜RT J i · sj H= iσ

−



ij

Di (ni · J i )2 .

i

i

(9.77)

i

Here the first three terms denote the d-band Hubbard model (see (1.51)). εi , tij , and Ui denote the atomic level on site i, the transfer integrals between sites i and j , and the intraatomic Coulomb interaction between d electrons on site i, respectively. The fourth term at the r.h.s. of (9.77) expresses the exchange couplings between the RE moment J i and the TM spin s i . The last term represents random anisotropy. The spin fluctuation theories of amorphous alloys based on the Hamiltonian (9.77) are left for future investigations.

Appendix A

Equivalence of the CPA Equations (3.83), (3.85), and (3.89)

We verify here the equivalence of three equations on the CPA, (3.83), (3.85), and (3.89). We start from the CPA equation (3.83) showing the disappearance of the single-site t matrix in average.   δviσ t˜iσ  = = 0. (A.1) 1 − δviσ Fiσ Here δviσ = viσ − Σiσ (z), viσ and Σiσ (z) being the impurity potential and the coherent potential, respectively. Note that we have generalized the coherent potential to be site-dependent. Fiσ (z) is the coherent Green function defined by   Fiσ (z) = (z − Σσ − t)−1 ii . (A.2) The average ∼ in (A.1) is a classical average defined by (3.84). Since the t-matrix in (A.1) is written as   −1 Fiσ −1 − 1 Fiσ , t˜iσ = −1 Fiσ − δviσ

(A.3)

we obtain an alternative expression for the CPA condition as (i)

Giσ (z) = Fiσ (z).

(A.4) (i)

This is identical with the CPA equation (3.85), and the Green function Giσ (z) is defined by (i)

Giσ (z) =

1 −1 Fiσ (z) − δviσ

.

(A.5)

(i)

The Green function Giσ (z) is the on-site Green function for an impurity embedded in the effective medium, of which the Hamiltonian is given by (i)

  Hσ (z) j l = viσ (z, ξ )δij + Σσ (z)(1 − δij ) δj l + tj l (1 − δj l ). (A.6) Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

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A Equivalence of the CPA Equations (3.83), (3.85), and (3.89)

(i) To verify this fact, we rewrite the Hamiltonian as Hσ (z) = H˜ σ (z) + δviσ . ˜ ˜ Here Hσ (z) is the coherent Hamiltonian defined by (Hσ (z))j l = Σj σ (z)δj l + tj l (1 − δj l ). δviσ is defined by (δviσ )j l = (viσ − Σiσ (z))δij δj l . We then expand the Green function on the impurity site as 

−1    ˜ σ (z) + G ˜ σ (z)viσ (z)G ˜ σ (z) + · · · . z − Hσ(i) (z) = G (A.7) ii ii

˜ σ (z) is the resolvent of the Hamiltonian H˜ σ (z), i.e., G ˜ σ (z) = (z − H˜ σ (z))−1 . Here G Equation (A.7) means that 

−1  ˜ iiσ (1 − δviσ G ˜ iiσ )−1 . z − Hσ(i) (z) =G (A.8) ii ˜ iiσ (z), we find the relation: Since Fiσ = G 

−1  (i) . Giσ (z) = z − Hσ(i) (z) ii

(A.9)

The CPA equations (A.1) and (A.4) are expressed as the stationary condition for the free energy (3.82):

  βU −1 ˜ FSSA = F − β ln (A.10) dξi e−βEi (ξi ) . 4π i

Here the coherent part of the free energy F˜ is defined by (3.72):     1 F˜ = dω f (ω) Im tr ln(z − Σ − t) . π σ

(A.11)

Note that tr at the r.h.s denotes the trace over sites. The single-site energy Ei (ξi ) is given by (3.78):  

1 1 Ei (ξi ) = dω f (ω) Im ln z − δviσ (z)Fiσ (z) + U ξi2 − ζi2 . (A.12) π 4 σ To derive the stationary condition, we rewrite the free energy (A.10) as follows.   βU ˜  −1 dξj e−β(F + i Ei (ξi )) . (A.13) FSSA = −β ln 4π j

Here F˜ +



 Ei (ξi ) =

i

dω f (ω)



1 1 2 Im Xσ (z) + Ui ξi − ζi2 , (A.14) π 4 σ i

and Xσ (z) is defined by Xσ (z) = tr ln(z − Σσ − t) +

   ln z − δviσ (z, ξ )Fiσ (z) . i

(A.15)

A Equivalence of the CPA Equations (3.83), (3.85), and (3.89)

303

Taking the variation of FSSA with respect to the coherent potential Σiσ (z), we have       1 Ei (ξi ) = dω f (ω) Im δXσ (z). (A.16) δFSSA = δ F˜ + π σ i

Note that we have used here the stationary condition for ζi (ξ ); ∂Ei (ξ )/∂ζi (ξ ) = 0. The variation of Xσ (z) is obtained as  

−1 z − δviσ (z, ξ )Fiσ (z) Fiσ (z)δΣiσ (z) Fiσ (z)δΣiσ (z) + δXσ (z) = − i

i



−1 − z − δviσ (z, ξ )Fiσ (z) δviσ (z, ξ )δFiσ (z).

(A.17)

i

Thus the average is obtained as follows    (i) δXσ (z) = t˜iσ (z)δFiσ (z). (A.18) −Fiσ (z) + Giσ (z) δΣiσ (z) − i

i

By making use of the CPA equations (A.1) and (A.4), we find the stationary condition as follows. δXσ (z) = 0.

(A.19)

From (A.16) and (A.19), we obtain the stationary condition. δFSSA = 0. δΣiσ (z)

(A.20)

This is identical with (3.89), so that we have verified that (3.83), (3.85), and (3.89) are equivalent to each other.

Appendix B

Dynamical CPA Based on the Multiple Scattering Theory

The dynamical CPA theory presented in Sect. 3.4 is also obtained by using the multiple scattering theory in the disordered alloys [51]. We derive in this appendix the dynamical CPA equation on the basis of the temperature Green function and the multiple scattering theory. The temperature Green function Gij σ (τ − τ  ) is given by an average of the timedependent Green function Gij σ (τ, τ  ) with respect to the energy functional E[ξ, η] as shown in (3.48):   δξi δηi Gij σ (τ, τ  )e−βE[ξ,η]



 i Gij σ τ − τ  = Gij σ τ, τ  =   . (B.1) −βE[ξ,η] δξi δηi e i

The time-dependent Green function is determined by solving the Dyson equation (3.43):  β 









  Gij σ τ, τ = gij σ τ − τ + dτ  gikσ τ − τ  vkσ τ  Gkj σ τ  , τ  . 0

k

(B.2) Here gij σ (τ − τ  ) is the Green function for noninteracting Hamiltonian H0 , and viσ (τ ) is the time-dependent random potential given by (3.37). The Dyson equation is written in matrix form as

−1 G = g + gvG = g −1 − v . (B.3) The matrices are for example defined as (G)iτ σj τ  σ  = Gij σ (τ, τ  )δσ σ  . Now, we consider a scattering from an effective potential Σ  . Inserting potential  Σ into the potential part in the above expression, we expand the Green function with respect to the scattering potential δv = v − Σ  as ˜ + Gδv ˜ G ˜ + Gδv ˜ Gδv ˜ G ˜ + ··· . G=G Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

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B Dynamical CPA Based on the Multiple Scattering Theory

˜ is the coherent Green function defined by Here G

˜ = g −1 − Σ  −1 . G

(B.5)

The Green function (B.4) is expressed with use of the T matrix as ˜ + GT ˜ G. ˜ G=G

(B.6)

Here the T matrix is defined by the potential scattering as ˜ + · · · = δv(1 + GT ˜ ). T = δv + δv Gδv

(B.7)

On the other hand the real self-energy Σ for the temperature Green function (B.1) is defined by

−1 . G = G = g −1 − Σ

(B.8)

This is nothing but the Dyson equation for the temperature Green function. To obtain the expression of Σ with use of the T matrix, we solve the above equation as Σ = ˜ −1 (1 + GT ˜ g −1 − G−1 , and substitute the expression G−1 = G )−1 , which is obtained from (B.6), into G at the r.h.s., so that we obtain

−1 ˜ ˜ −1 1 + GT Σ = g −1 − G  .

(B.9)

According to the definition of the coherent Green function (B.5), we have the rela˜ −1 . Thus we can express the self-energy (B.9) as tion g −1 = Σ  + G

−1 ˜  . Σ = Σ  + T  1 + GT

(B.10)

This is the exact self-energy expression when we start from an effective medium Σ  . The correction is given by the T matrix caused by the scattering potential δv. In order to obtain the averaged T matrix, i.e., T  at the r.h.s. of (B.9), we solve (B.7) with respect to T as ˜ −1 δv. T = (1 − δv G)

(B.11)

˜ = F +F  ˜ into the diagonal part F and the off-diagonal part F  ; G Next, we divide G to make the single-site approximation (SSA). Substituting the expression into (B.11) and expanding T with respect to F  , we obtain

−1 T = 1 − t˜F  t˜,

(B.12)

and t˜ is given by t˜ = (1 − δvF )−1 δv =

 i

t˜i .

(B.13)

B Dynamical CPA Based on the Multiple Scattering Theory

307

Here t˜i denotes the single-site t matrix on site i, which is given by t˜i = (1 − δvi Fi )−1 δvi ,

(B.14)

˜ ii . and Fi = G Making the SSA we obtain T  from (B.12) as T  ≈ t˜.

(B.15)

˜ ≈ F in Substituting (B.15) into (B.10) and making the single-site approximation G the denominator, we find the self-energy in the SSA as

−1 Σ = Σ  + 1 + t˜F t˜,

(B.16)

or Σ = Σ +



−1 1 + t˜i Fi t˜i .

(B.17)

i

Note that the second term depends on the effective medium Σ  . When we choose = v (i.e., the Hartree–Fock value), we call (B.17) the average t-matrix approximation (ATA). The best choice of the effective medium Σ  should be obtained from the condition: Σ

t˜i  = (1 − δvi Fi )−1 δvi  = 0.

(B.18)

This is identical with the dynamical CPA equation (3.112). The present method indicates that the medium Σ  = Σ is the self-energy of the temperature Green function. Equation (B.17) is often used to obtain the CPA solution. Assuming that we start from the Hartree–Fock value Σ  = v, we calculate the corrections according to the second term at the r.h.s. of (B.17). If the obtained self-energy Σ (1) does not agree with Σ  , we renew as Σ  = Σ (1) and repeat the procedure until the selfconsistency Σ = Σ  is achieved. Equation (B.17) is identical with (3.97) when iωl in the frequency representation is replaced by z = ω + iδ.

Appendix C

Derivation of the Single-Site Spin Fluctuation Theory from the Dynamical CPA

We verify in this Appendix that the dynamical CPA presented in Sect. 3.4 reduces to the single-site spin fluctuation theory (SSF) given in Sect. 3.3 in the static limit. The free energy per atom in the dynamical CPA is given by (3.148) in the static approximation:  βU −1 dξ e−βEi (ξ ) . (C.1) FCPA = F˜ − β ln 4π The static energy potential Ei (ξ ) is given by (3.149). Ei (ξ ) = −β −1



1

ln 1 − δviσ (iωl , ξ )Fiσ (iωl ) + U ξ 2 − ζi2 (ξ ) . 4

(C.2)

lσ

Here δviσ (iωl , ξ ) is defined by (3.79), the coherent Green function Fiσ (iωl ) is given by (3.88), and ζi (ξ ) by (3.150): ζi (ξ ) =

1  (i) Giσ (iωl , ξ ). β

(C.3)

lσ

The static Green function Giσ (z, ξ ) is defined by (3.86); Giσ (z, ξ ) = 1/(Fiσ (z)−1 − δviσ (z, ξ )). The dynamical CPA equation to determine the coherent potential is given by (3.143): (i)

(i)

(i)

Giσ (iωl , ξ ) = Fiσ (iωl ).

(C.4)

The CPA equation on the real axis is obtained by direct replacement iωl → z (= ω + iδ) as follows. G(i) iσ (z, ξ ) = Fiσ (z).

(C.5)

The frequency sums in (C.2) and (C.3) can also be transformed into the integrals on the real axis of the complex z plane. To prove this we adopt the Lehmann represenY. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

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310

C Derivation of the Single-Site Spin Fluctuation Theory

tation of the Green function.



(i)

Giσ (iωl , ξ ) =

ρiσ (ω, ξ ) dω . iωl − ω

(C.6)

Here ρiσ (ω, ξ ) is the single-particle density of states of interacting electrons, and is given by ρiσ (ω, ξ ) = −

1 Im G(i) iσ (ω + iδ, ξ ). π

Substituting (C.6) into (C.3), we find  1  eiωl η . dω ρiσ (ω, ξ ) ζi (ξ ) = β iωl − ω σ

(C.7)

(C.8)

l

Here we explicitly wrote a convergence factor with an infinitesimal positive number η. Using the following formula [43] on the Fermi distribution function at the r.h.s. of (C.8), f (ω) =

1 1  eiωl η = , eβω + 1 β iωl − ω

(C.9)

l

we find the expression with use of the Green function on the real axis.   ζi (ξ ) = dω f (ω) ρiσ (ω, ξ ).

(C.10)

σ

Next, we rewrite the energy potential (C.2). In this case, we consider the following function. 

1 eηz ln 1 − δviσ (z, ξ )Fiσ (z) . Φσ = dz βz (C.11) 2πi C e +1 Here contour C is given in Fig. C.1. Note that the same contour was used to prove the formula (C.9) [43]. Counting the residues along the imaginary axis, we find Φσ = −

∞

1  iωl η e ln 1 − δviσ (iωl , ξ )Fiσ (iωl ) . β

(C.12)

l=−∞

Next, we change the contour in (C.11) from C to C  + Γ1 + Γ2 in Fig. C.1, assuming that the integrand is analytic except for the real and imaginary axes, and evaluate the contribution from each path. We find that the contribution from the paths Γ1 and Γ2 vanishes, and only the contribution from C  remains, so that we obtain  ∞

1 Φσ = dω Im ln 1 − δviσ (z, ξ )Fiσ (z) . (C.13) π −∞

C Derivation of the Single-Site Spin Fluctuation Theory

311

Fig. C.1 Contour for calculation of frequency sums on the complex plane

Using (C.2), (C.12), and (C.13), we find the expression which is written by the quantities in the real axis.  ∞

1

1 Ei (ξ ) = dω Im ln 1 − δviσ (z, ξ )Fiσ (z) + U ξ 2 − ζi2 (ξ ) . (C.14) π 4 −∞ σ The equations (C.1), (C.14), (C.10), and (C.5) are identical with (3.82), (3.78), (3.91), and (3.85) in the SSF, so that we have verified that the dynamical CPA in the static approximation reduces to the SSF.

Appendix D

Expansion of Dνσ with Respect to Dynamical Potential

The determinant Dνσ in the harmonic approximation is defined by (3.165):

  Dνσ = det δlm − v˜σ (ν)δl−m,ν + v˜σ (−ν)δl−m,−ν g˜ σ (m) .

(D.1)

In this appendix, we derive an alternative expression of the determinant Dνσ which is expanded with respect to the dynamical potentials v˜σ (±ν), i.e., (3.167). The determinant Dνσ has the following form.   1 0 0   0 1 0   · · ·   0

 b1 0 Q {bi }{ci } =  0 b2 0   0 0 b3  ···   ···

··· ···

0 0

c1 0

0 c2

···

1 0

0 1

0 0

··· 0 ··· ··· · · · cν+1 ··· 0

0 0

1 0

··· 1

··· 0 ··· ···

0 .. .

0 cν+2

         · · ·  . . .  .     (D.2)

Application of the Laplace expansion to the first column of Q yields the relation,

Q {bi }{ci } = Q(N −1) − b1 c1 Q(N −2) .

(D.3)

Here Q(N −1) (Q(N −2) ) is the (N − 1) × (N − 1) ((N − 2) × (N − 2)) matrix with the same structure as Q, assuming that Q is the N × N matrix. Repeating the same expansions for Q(N −1) and Q(N −2) , we find that Q is the function of {bi ci }. Thus we have the relation



Q {bi }{ci } = Q {bi ci }{1} . Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

(D.4) 313

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D Expansion of Dνσ with Respect to Dynamical Potential

Thus we can consider Q({bi }{1}) as follows.   1 0 0 ··· 0   0 1 0 ··· 0   · · ·   0 1

 b1 0 Q {bi }{1} =  0 b2 0 · · · 0   0 0 b3 · · · 0  ··· ··· ···   ···

0 1

0 ··· 1 0 ··· 0 ··· 0 ···

0 0

1 ··· 0 1

1 0

··· 1 0 0 .. .

0 1

         · · ·  . . .  .    

(D.5)

We move the b1 row to the 2nd row, the bν+1 row to the 3rd row, the b2ν+1 row to the 4th row, and so on. Next, we move the bν+1 column to the 2nd column, the b2ν+1 column to the 3rd column, the b3ν+1 column to the 4th column, and so on. We have then

Q {bi }{1}     1 1 0 ···     b1 1 1 0 ···     0 bν+1 1 1 0 ···    ··· 1 0 ··· 0 b2ν+1 1     ··· ···     0 ··· ··· 0 1 1 0 ··· = .   0 1 1 0 ··· · · · 0 b2    0 1 1 · · ·  ··· ··· 0 0 bν+2   0 b2ν+2 0 · · ·     ··· ···     ..  ··· . (D.6) Thus the determinant Q is given by the products of the sub-determinants Applying the properties of the determinant Q to Dνσ , we find that Dνσ is written by the product of tridiagonal matrices as follows.   ..  .      Dνσ (k) =       

Dνσ = Dνσ (0)Dνσ (1) · · · Dνσ (ν − 1),

1 a−ν+kσ (ν)

1 1 akσ (ν)

0 1 1 aν+kσ (ν)

0

1 1 a2ν+kσ (ν)

        .  1    . .  .

(D.7)

(D.8)

D Expansion of Dνσ with Respect to Dynamical Potential

315

Here Dνσ (k) is the determinant of the tridiagonal matrix consisting of the Green functions with frequency remainder k for modulus ν, and anσ (ν) is defined by anσ (ν) = v˜σ (ν)v˜σ (−ν)g˜ σ (n − ν)g˜ σ (n).

(D.9)

Here we used the notation v˜σ (ν) = v˜σ (iων ) and g˜ σ (n) = g˜ σ (iωn ), for simplicity. Note that according to the Laplace expansion theorem Dνσ (k) is expressed by the determinants of the sub-matrices as (0) (0) (1) (1) Dνσ (k) = Dνσ (ν, k)Dνσ (−ν, k) − akσ (ν)Dνσ (ν, k)Dνσ (−ν, k),   1 1 0   a(m+1)ν  +kσ (ν) 1 1  1 1 a(m+2)ν  +kσ (ν)

 (m)  Dνσ ν , k =   +kσ (ν) 1 a (m+3)ν      0

(D.10)        1  . . .  .  (D.11)

(m) Furthermore the determinant Dνσ (±ν, k) is expanded as follows. (m) Dνσ (±ν, k) =

∞ 

n (m) (−)n v˜σ (ν)v˜σ (−ν) An±νkσ .

(D.12)

n=0

Here (m)

A0±νkσ ≡ 1, (m)

An±νkσ =

∞ 

l 1 −2

l1 =m+1 l2 =m+1

(D.13)

ln−1 −2

···



aˆ l1 (±ν)+kσ (ν)aˆ l2 (±ν)+kσ (ν) · · · aˆ ln (±ν)+kσ (ν),

ln =m+1

(D.14) and aˆ nσ (ν) = g˜ σ (n − ν)g˜ σ (n).

(D.15)

To prove (D.12), let us consider the determinant of a finite matrix as follows.     1 1 0     a(m+1)ν+k 1 1     1 1 a(m+2)ν+k     (m,M) a(m+3)ν+k 1 1 D =  . (D.16)   . .   .     0    aMν+k 1 

316

D

Expansion of Dνσ with Respect to Dynamical Potential

By applying the Laplace expansion, we have the recursion relation as D (m,M) = D (m,M−1) − aMν+k D (m,,M−2) .

(D.17)

In the same way, we obtain the following recursion relations in general. D (m,n) = D (m,n−1) − anν+k D (m,n−2) ,

(D.18)

D (m,m+1) = 1 − a(m+1)ν+k ,

(D.19)

D (m,m) = 1.

(D.20)

Now, we start from the following identity. D (m,M) = 1 +

M  (m,l )

D 1 − D (m,l1 −1) .

(D.21)

l1 =m+1

Substituting (D.18) into (D.21), we obtain D (m,M) = 1 −

M 

al1 ν+k D (m,l1 −2) .

(D.22)

l1 =m+1

Repeating the same procedure for D (m,l1 −2) , we have   l M 1 −2  al1 ν+k 1 − al2 ν+k D (m,l2 −2) , D (m,M) = 1 − l1 =m+1

(D.23)

l2 =m+1

and finally we reach the following expansion. D

(m,M)

ln−1 −2 l M 1 −2   n =1+ (−) ··· al1 ν+k · · · aln ν+k . n=1

l2 =m+1

(D.24)

ln =m+1

Substituting an = v˜σ (ν)v˜σ (−ν)g˜ σ (n − ν)g˜ σ (n) into (D.24) and taking the limit M → ∞, we obtain (D.12). Substituting (D.12) for m = 0 and 1 into (D.10), we obtain an expression of Dνσ (k) which is expanded with respect to the dynamical potential as follows. Dνσ (k) =

  ∞  1 iβ v˜σ (ν)v˜σ (−ν) l (l) Bνσ (k). l! 2πν

(D.25)

l=0

Here (0) Bνσ (k) ≡ 1,

(D.26)

D Expansion of Dνσ with Respect to Dynamical Potential

 (l) Bνσ (k) = (−)l l!

 ×

2πν iβ

(0) Alνkσ

317

l

 l−1  (0)

(0) (1) (1) + Amνkσ Al−m−νkσ + aˆ kσ Amνkσ Al−1−m−νkσ . (D.27) m=0

From (D.7) and (D.25) we obtain the expansion form of Dνσ as follows. Dνσ

 l ∞ 

l i 1 (l) 4β v˜ σ (ν)v˜σ (−ν) = Bνσ , l! 8πν l=0

(l) = Bνσ

 ν−1

k=0 lk

l!

=l

$ [ ν−1 k=0 lk !]

ν−1 

(D.28)

 (lk ) Bνσ (k) .

(D.29)

k=0 (l)

Equation (D.28) is identical with (3.167); the coefficient Bνσ is given by (D.29).

Appendix E

Linear Response Theory

We derive in this appendix the linear response formula (4.53) for the time dependent perturbation [83], which was used in the calculation of the dynamical susceptibility in Sect. 4.3.  t



B(t) = χBA t − t  F t  dt  . (E.1) −∞

Here F (t) is a time-dependent external force, B(t) denotes the linear change of a physical quantity B, and χBA (t − t  ) is the linear response function. Let us consider the ensemble in which each system was in the equilibrium state at t = −∞ and assume that it develops according to the Hamiltonian H of the system, neglecting a small perturbation from the outside during the time development. The average of a physical quantity Bˆ is then given by    ˆ = ˆ α (t) . B ρα Ψα (t)|B|Ψ (E.2) α

Here ρα is the distribution in the equilibrium state at the beginning. The state Ψα (t) develops according to the Schrödinger equation. The time-dependent average (E.2) is expressed as

ˆ = tr ρ(t)Bˆ . B (E.3) Here ρ(t) is called the density matrix which is defined by     Ψα (t) ρα Ψα (t). ρ(t) =

(E.4)

α

The time development of the density matrix operator ρ(t) is obtained from the Schrödinger equation as follows. dρ(t) 1 = [H, ρ]. dt i Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

(E.5) 319

320

E

Linear Response Theory

Assume now that the system is perturbed by a time-dependent force F (t), whose interaction is given by ˆ (t). H1 (t) = −AF

(E.6)

Here Aˆ is an operator of a physical quantity and the force F (t) is assumed to be F (−∞) = 0. Accordingly we assumed that the Hamiltonian H consists of the original Hamiltonian H0 and H1 (t): H = H0 + H1 (t). The density matrix is written by the distribution in the equilibrium state ρ0 and the deviation ρ(t) as follows. ρ(t) = ρ0 + ρ(t). With use of ρ(t) the change of physical quantity Bˆ is given as

ˆ ˆ 0 = tr ρ(t)Bˆ . B(t) ≡ B(t) − B

(E.7)

(E.8)

ˆ 0 denotes the thermal average of Bˆ in the equilibrium state which is deterHere B mined by the Hamiltonian H0 . The density matrix ρ(t) is obtained from the equation of motion (E.5). By making use of the relation [H0 , ρ0 ] = 0 and linearizing the equation of motion with respect to the perturbation, we obtain  dρ(t) 1 1 ˆ H0 , ρ(t) . = − [A, ρ0 ]F (t) + dt i i

(E.9)

Integrating the above equation from −∞ to t and solving the equation iteratively, we find that 

1 1 1 t  ˆ ρ0 ]e− i H0 (t−t  ) F t  . (E.10) dt  e i H0 (t−t ) [A, ρ(t) = − i −∞ Here we used the following formula for any operators S and T . eS T e−S = T + [S, T ] +

 1 S, [S, T ] + · · · . 2!

Substituting (E.10) into (E.8), we obtain the linear response formula.  t



B(t) = χBA t − t  F t  dt  . −∞

(E.11)

(E.12)

Here the linear response function χBA (t) is given by a time correlation function at equilibrium state as follows. i ˆ χBA (t) = [Bˆ H (t), A]. 

(E.13)

Here we have omitted the suffix 0 in the average   for simplicity promising that the average is taken in the equilibrium state. Accordingly, we have redefined H by

E

Linear Response Theory

321

the Hamiltonian H0 in the equilibrium state. Bˆ H (t) is the Heisenberg representation ˆ i.e., Bˆ H (t) = exp(iH t/)Bˆ exp(−iH t/). Equations (E.1) of physical quantity B, and (E.13) are identical with (4.53) and (4.54) in Sect. 4.3.

Appendix F

Isothermal Molecular Dynamics and Canonical Distribution

We adopted in Sect. 6.3 the isothermal molecular dynamics method [106], and solved the following equations of motion (6.64), (6.65), and (6.66) for a given temperature T : ξ˙iα =

piα , μLM

∂Ψ (ξ ) − ηα · piα , ∂ξiα   2 1  piα − NT . η˙ α = Q μLM

p˙ iα = −

(F.1) (F.2) (F.3)

i

Here μLM , ξ i , and pi denote the mass, the position vector, and the momentum for a fictitious particle i, respectively. Ψ (ξ ) is the potential energy between particles, so that −∂Ψ (ξ )/∂ξiα denotes the force acting on the particle i. ηα are the friction variables, and Q is a constant. Moreover N is the number of particles of the system. In this Appendix we will prove that the equations of motion (F.1), (F.2), and (F.3) yield the canonical ensemble whose distribution in the {ξ , p} space is given by f (ξ , p) = Ce−βH (ξ ,p) .

(F.4)

Here β is the inverse temperature and C is a constant. H (ξ , p) is the Hamiltonian for a classical system with the interaction potential Ψ (ξ ). H (ξ , p) =

 p2 i + Ψ (ξ ). 2μLM

(F.5)

i

Let us consider the ensemble of the system described by (F.1), (F.2), and (F.3) and introduce the probability distribution f˜(ξ , p, η, t) in the 6N + 3 phase space. Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

323

324

F Isothermal Molecular Dynamics and Canonical Distribution

The latter satisfies the equation of continuity as follows. ∂ f˜ ∂ ˜˙ ∂ ˜ ∂ ˜ ˙ + ˙ = 0. + (f ξ ) + (f p) (f η) ∂t ∂ξ ∂p ∂η

(F.6)

  ∂ ∂ d f˜ ∂ ˙ ξ+ p˙ + η˙ f˜. =− dt ∂ξ ∂p ∂η

(F.7)

Alternatively,

Substituting the equations of motion (F.1), (F.2), and (F.3) into (F.7), we find   d f˜ =N ηα f˜. (F.8) dt α On the other hand, we consider the extended Hamiltonian defined by 1 Qηα2 . H˜ (ξ , p, η) = H (ξ , p) + 2 α

(F.9)

Taking the derivative of H˜ (ξ , p, η) with respect to t and making use of the equations of motion (F.1), (F.2), and (F.3), we obtain   d H˜ (ξ , p, η) (F.10) ηα . = −N T dt α From (F.8) and (F.10), we find the relation d f˜ 1 d H˜ ˜ =− f. dt T dt

(F.11)

˜ −β H˜ (ξ ,p,η) . f˜(ξ , p, η) = Ce

(F.12)

Solving the equation, we obtain

Here β = 1/T and C˜ is a constant. Thus the distribution f (ξ , p) is given by the canonical distribution as follows.  f (ξ , p) = f˜(ξ , p, η) dη = Ce−βH (ξ ,p) . (F.13) The ergodic theorem tells us that the average of a physical quantity A(ξ (t), p(t)) is given by  

1 t0 A ξ (t), p(t) dt. (F.14) A(ξ , p)f (ξ , p) dξ dp = lim t0 →∞ t0 0 Thus we can calculate thermal average by taking the time average of the physical quantity after we solve the equations of motion (F.1), (F.2), and (F.3). In the case of magnetic moment (6.60), we obtain the thermal average from time average (6.63).

Appendix G

Recursion Method for Electronic Structure Calculations

The recursion method allows us to calculate the Green function for the tight-binding Hamiltonian without translational symmetry. Let us assume that the Hamiltonian is given by Hij = εi δij + tij (1 − δij ).

(G.1)

Here εi is the atomic level on site i and tij is the transfer integral between sites i and j . The Green function Gij (z) (= (G(z))ij ) is defined by   Gij (z) = (z − H )−1 ij .

(G.2)

Here (H )ij = Hij denotes the Hamiltonian matrix. Now, we make a unitary transformation of the Hamiltonian matrix H to a tridiagonal Hamiltonian matrix H  .

H  = U † H U = u†i H uj . (G.3) Here U = (uij ) = (u1 , u2 , . . . , uN ) is a unitary matrix such that U † U = 1 (or u†i uj = δij ). The unitary column vectors {uj } are defined by (uj )i = uij (i = 1, . . . , N). The Green function G (z) = [(z − H  )−1 ]ij for the Hamiltonian H  is connected with the original Green function matrix G(z) as u†i G(z)uj = Gij (z).

(G.4)

In particular, we have the following relation when i = j = 1 and the unit vector u1 is chosen such that (u1 )i = 1 and (u1 )m = 0 (m = i). Gii (z) = Gii (z).

(G.5)

The Hamiltonian is tridiagonalized by the Lanczos method. There we produce both the unitary vectors and the tridiagonal matrix elements with use of the recursive relations. We choose the starting vector to be a unit vector, and produce a new Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

325

326

G

Recursion Method for Electronic Structure Calculations

orthogonal basis set {ui } successively with use of the following relations. b1 u2 = H u1 − a1 u1 ,

(G.6)

bn un+1 = H un − an un − bn−1 un−1 .

(G.7)

The unit vector un+1 is obtained by normalization of the r.h.s. of (G.6) or (G.7). The coefficients an and bn are also successively obtained by using the orthogonality u†i uj = δij . an = u†n H un ,

(G.8)

bn = un+1 H un .

(G.9)

∗ , and u† H u = 0 for l < n − 1 Note that u†l H u1 = 0 for l > 2, u†n−1 H un = bn−1 n l and l > n + 1; thus the Hamiltonian Hij = u†i H uj is tridiagonalized as follows.

⎛

a1 ⎜b1∗ ⎜ ⎜ ⎜0 ⎜  H =⎜ ⎜0 ⎜ ⎜ .. ⎝.

b1 a2

0 b2

··· 0

···

b2∗

a3 .. .

b3 .. .

0 .. .

0

···

0

∗ bN −2 0

0 0 .. .

aN −1 ∗ bN −1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ bN −1 ⎠ aN

(G.10)

Choosing u1 so that (u1 )i = 1 and (u1 )m = 0 (m = i), we can obtain the diagonal Green function Gii (z) via (G.5). The Green function Gii (z) is expressed by using Cramer’s formula and Laplace’s expansion as Gii (z) = D1 /D0 = 1/(z − a1 − |b1 |2 D2 /D1 ). Here D0 = det(z − H  ), D1 is the cofactor of (z − H  )11 , and D2 is the cofactor of (z − H  )22 for the submatrix of D1 . Repeating the same procedure, we find that the Green function Gii (z) is given by a continued fraction as follows. 1

Gii (z) = Gii (z) =

.

|b1 |2

z − a1 −

|b2 |2 ..

z − a2 − ... ... −

(G.11)

.

|bn−1 |2 z − an − Tn (z)

Here Tn (z) is a terminator at the n-th order recursion. The square-root terminator with use of the asymptotic recursion coefficients a∞ and b∞ is usually applied. ' ) 1( (G.12) Tn ≈ T∞ = z − a∞ − (z − a∞ )2 − 4|b∞ |2 . 2

G Recursion Method for Electronic Structure Calculations

327

The projected density of states is obtained as ρi (ω) = −

1 Im Gii (z). π

(G.13)

In some cases, we need the off-diagonal Green function √ Gij (z). In this case √ we start from the unit vectors u1± such that (u1± )i = 1/ 2, (u1± )j = ±1/ 2, and (u1± )m = 0 (m = i, j ). We have then the relation Gij + Gj i = u†1+ Gu1+ − √ u†1− Gu1− . When we start from the unit vectors v 1± such that (v 1± )i = 1/ 2, √ (v 1± )j = ±i/ 2, and (v 1± )m = 0 (m = i, j ), we have the relation Gij − Gj i = i(v †1− Gv 1− − v †1+ Gv 1+ ). Thus we obtain the off-diagonal Green function as Gij (z) =



 1  G (z) − G22 (z) − i G33 (z) − G44 (z) . 2 11

(G.14)

Here G11 (z) = u†1+ Gu1+ , G22 (z) = u†1− Gu1− , G33 (z) = v †1+ Gv 1+ , and G44 (z) = v †1− Gv 1− . These diagonal Green functions are obtained by the recursion method as presented above. In the MD calculation, we need to obtain the following off-diagonal Green functions to calculate the magnetic forces (see (6.70) and (6.71)).  (σx G)iLσ iLσ = GiL↑iL↓ + GiL↓iL↑ , (G.15) σ

 (σy G)iLσ iLσ = i(GiL↑iL↓ − GiL↓iL↑ ).

(G.16)

σ

√ To obtain (G.15),√we consider the unit vector u1± such that (u1± )iL↑ = 1/ 2, (u1± )iL↓ = ±1/ 2, and (u1± )j L σ = 0 (j = i). Then we obtain GiL↑iL↓ + GiL↓iL↑ = u†1+ Gu1+ − u†1− Gu1− . Each term of the r.h.s. is given by a continued fraction. In √ the case of (G.16), √ we consider the unit vector u1± such that (u1± )iL↑ = 1/ 2, (u1± )iL↓ = ±i/ 2, and (u1± )j L σ = 0 (j = i). We then obtain i(GiL↑iL↓ − GiL↓iL↑ ) = u†1+ Gu1+ − u†1− Gu1− .

Appendix H

An Integral in the RKKY Interaction

When we calculate the RKKY interaction in the free electron model, we employ the formula (7.32) for the integral (7.31) in Sect. 7.1. We derive in this appendix the formula (7.32): I (R) =

iπ (−2kF R cos 2kF R + sin 2kF R). kF R 3

(H.1)

The integral I (R) is defined by  q I (R) = eiqR . dq qF 2kF −∞ 

∞



(H.2)

Here the function F (x) is given by (7.27):   1 − x 2  1 + x  . ln F (x) = 1 + 2x 1−x

(H.3)

Note that F (x) is an even function, F (0) = 2, and F (x) → 2/3x 2 (x → ∞). By integration by parts, we can rewrite the integral (H.2) as follows.  ∞ 1 d[(q/2kF )F (q/2kF )] iqR e . I (R) = − dq (H.4) iR −∞ d(q/2kF ) Here d[xF (x)]/dx = 2 − x ln |1 + x|/|1 − x|. Making use of the integration by parts again, we obtain      ∞  2kF + q  1 iqR  + 4kF q  I (R) = dq ln (H.5)  2k − q  4k 2 − q 2 e . 2kF R 2 −∞ F F The first term at the r.h.s. of (H.5) is expressed by integration by parts as    ∞   2kF + q  iqR 4kF i ∞ eiqR e dq ln = dq 2 .  2kF − q R −∞ 4kF − q 2 −∞ Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

(H.6) 329

330

H An Integral in the RKKY Interaction

Fig. H.1 Contour on the complex plane to calculate the integrals (H.8) and (H.11)

Thus we obtain I (R) =



2 iR 3

∞

−∞

dq

eiqR 2 − 2 2 2 q − 4kF R



∞

−∞

dq

qeiqR . q 2 − 4kF2

Let us calculate the first term at the r.h.s. of (H.7):  ∞ eiqR A= dq . q 2 − 4kF2 −∞

(H.7)

(H.8)

In order to calculate the integral A, we consider the following integral along the contour on the complex plane as shown in Fig. H.1, which vanishes according to the Cauchy theorem.  A+

C1 +C2 +C3

dq

eiqR = 0. q 2 − 4kF2

(H.9)

Here the integral along C1 (C2 ) is obtained as (iπ/4kF ) exp(−i2kF R) ((−iπ/4kF )× exp(−i2kF R)). The integral along C3 is proven to vanish. Thus we obtain A=−

π sin 2kF R . 2kF

(H.10)

In the same way, we obtain 

∞

−∞

dq

qeiqR = iπ cos 2kF R. q 2 − 4kF2

(H.11)

From (H.7), (H.10), and (H.11), we obtain I (R) =

iπ (−2kF R cos 2kF R + sin 2kF R), kF R 3

which is identical with (H.1), i.e., (7.32) in Sect. 7.1.

(H.12)

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Index

Symbols 3d transition metal, 18, 29 5-fold equivalent band model, 210 α-Mn, 149 γ -Mn, 24, 168 A Ab-initio MD method, 256 Action, 97 Alexander–Anderson–Moriya interaction, 226 Amorphous Co, 268 Amorphous Co1−x Bx , 254 Amorphous Fe, 267 Amorphous metals, 258 Amplitude of local moment, 78 Amplitudes of the magnetic moments, 175 Analytic continuation, 104 Anderson lattice Hamiltonian, 183 Anderson model, 183 Anisotropy, 170 Anti-bonding state, 11 Antibonding, 54 Antiferromagnetic insulators, 18 Antiferromagnetic states, 149 Antiferromagnetic structure, 24 Antiferromagnetism, 18, 244, 269 ASRO effects, 295 Atomic magnetic moment, 23 Atomic orbitals, 4 Atomic potential, 226 Atomic short-range order, 231 Atomic short-range order parameters, 281 Atomic sphere approximation, 57 Atomic-size difference, 284, 292 Au1−x Fex , 28 Au2 Mn, 26 Average coordination number, 281

Average t -matrix approximation, 307 Averaged t -matrix approximation, 79 Azimuthal quantum number, 4 B Band narrowing factor, 45 Band-narrowing, 36 Band-type AF, 162 BCS gap equation, 153 Bernal holes, 295 Bethe ansatz method, 201 Bethe approximation, 219, 261 Bethe lattice, 155 Binding energy, 299 Binomial distribution function, 221 Bloch theory, 206 Bohr magneton, 2 Bonding, 54 Bonding state, 11 Brillouin curve, 215, 237 Brillouin zone, 23, 152 Broken AF, 246 Broken antiferromagnetic states, 249 C Canonical distribution, 324 Canonical structure constant, 55 Car–Parrinello method, 256 Cauchy theorem, 330 Cavity Green function, 100 Ce, 201 Charge neutrality potential, 213 Chemical bondings, 296 Closed shell, 16 Cluster SG, 251, 272 Cluster spin glass, 247 Co, 23, 29, 60, 67, 112, 160

Y. Kakehashi, Modern Theory of Magnetism in Metals and Alloys, Springer Series in Solid-State Sciences 175, DOI 10.1007/978-3-642-33401-6, © Springer-Verlag Berlin Heidelberg 2012

339

340 Co–P, 295 Co–Y amorphous alloys, 291 Co-rich Co–Y amorphous alloys, 255 Co2 Y alloy, 292 Coc Y1−c , 293 Coexistence, 244, 247 Coherent action, 99 Coherent Green function, 75, 83, 85, 262 Coherent potential, 75, 76, 109, 154 Coherent potential approximation (CPA), 77, 206 Collinear, 247, 265 Collinear ferromagnetism, 266 Collinear SG, 266 Commensurate, 25 Commensurate multiple MSDW, 172 Common band model, 283 Complete ferromagnetism, 32 Complex magnetic structures, 244, 278 Configurational average, 27 Configurational disorders, 280 Conical magnetic structure, 26 Constraint LDA, 108 CoO, 17 Coulomb integral, 7 Coulomb interaction, 4, 34 Covalent bond, 12 Cowley’s atomic short-range order parameter, 221 CPA, 77, 207, 301 CPA equation, 77, 100, 102, 104, 208, 213 Cr, 18, 24, 25, 61, 149, 160 Cr–Mn, 150 Cr–V, 150 CrBr2 , 17 Critical Coulomb interaction, 46 CrO2 , 17, 26 Crystalline-field effects, 297 Cu1−x Mnx , 28 Cu3 Au, 231 Cubic harmonics, 5 Curie constant, 11 Curie law, 11, 195 Curie temperature, 66, 111, 139, 269, 274 Curie–Weiss law, 66, 112, 113, 146, 171 Curie-temperature Slater–Pauling curves, 241 Cut-off frequency, 123 D Debye cut-off frequency, 153 Decoupling approximation, 225, 263 Degree of structural disorder, 277 Dense random packing of hard spheres (DRPHS), 256

Index Density correlations, 47 Density functional theory, 49, 105, 298 Density of states, 15 Determines, 163 Diagonal coherent Green function, 103 Diamagnetism, 4 Direct exchange interaction, 20, 181 Distribution, 263 Distribution function, 227, 285 Distribution function method, 227 DMFT, 89, 96, 102 Double Q MSDW, 173 Double Q multiple SDW, 27 Double counting, 7 Double occupancy, 12, 31 Double occupation number, 36, 45 Dy, 26 Dy–Co, 297 Dy–Fe, 297 Dynamical coherent-potential approximation, 83 Dynamical CPA, 305, 309 Dynamical CPA equation, 85, 88, 110, 307 Dynamical mean-field theory, 89, 96 Dynamical potential, 90, 313 Dynamical spin fluctuations, 83 Dynamical susceptibility, 126, 129, 143 Dyson equation, 71, 85, 97, 100, 219 Dzyaloshinskii–Moriya interaction, 180 E Effective action, 97 Effective Bohr magneton number, 81, 112, 114 Effective Coulomb interaction, 37, 40, 46 Effective exchange interaction, 35 Effective Hamiltonian, 20, 95 Effective locator, 259 Effective mass, 46 Effective medium, 224, 262 Effective potential, 91, 109 Electron correlations, 36 Electron gas, 52 Electron hopping integral, 12 Electron–electron attractive interaction, 153 Electron–electron interaction, 4 Electron–phonon interaction, 153 Electrostatic potential, 7 Energy, 79 Energy dependent effective Liouville operator, 103 Energy functional, 70 Entropy, 79 Envelope function, 55 Equation of motion method, 129

Index Equations of motion, 103, 130, 323 Ergodic theorem, 324 Ergodicity, 166 Eu, 26 Exchange correlation energy, 50 Exchange energy parameter, 59 Exchange energy potential, 226 Exchange integral, 7 Exchange interaction, 34, 226 Exchange potential, 7 Exchange splitting, 137 Exchange-correlation potential, 50 F f-electron, 296 Fcc transition metals, 167 Fe, 23, 29, 35, 60, 61, 67, 81, 93, 111, 133, 160 Fe powder, 279 Fe–Co, 241 Fe–Cr, 240, 244 Fe–ET amorphous alloys, 279 Fe–metalloid alloys, 279 Fe–Ni, 233, 240, 242, 288 Fe–V, 240 Fe–Zr, 289 Fe-rich amorphous alloys, 254 Fe1−x Crx , 209 Fe100−x Cox , 179 Fe2 B, 295 Fe2 P, 295 Fe3 P, 295 Fe80 B20 , 295 Fec B1−c , 277 Fec Ni1−c , 215 Fec Y1−c , 293 Fex Cr100−x , 244 Fex Mn1−x , 179 FeO, 17 Fermi distribution function, 7, 32 Fermi level, 15 Fermi liquid theory, 201 Fermi surface, 161 Ferrimagnetism, 298 Ferromagnetic instability, 233, 235 Ferromagnetic insulators, 17 Ferromagnetic state, 191 Ferromagnetic structure, 23 Ferromagnetism, 29, 244, 265, 268, 278 Feynman diagram, 97 Field operator, 3 Finite-temperature theory of the LEE, 258 Finite-temperature theory of the local environment effects, 223 First-order transition, 215

341 First-principles dynamical CPA, 104 First-principles TB-LMTO Hamiltonian, 58 First-principles tight-binding, 105 Fluctuation–dissipation theorem, 128 Fourier representation, 87 Free electron model, 329 Free energy, 69, 86, 109, 164 Free spin-pairs, 248 Functional integral method, 68 G g-value, 2 Gap-type antiferromagnetism, 154 Gd, 29 Gd–Co, 298 Gd–Fe, 298 Gd–Ni, 298 Gd33 Fe67 , 298 GdFe2 , 299 Generalized gradient approximation, 62 Generalized Slater–Pauling curve, 254 Generalized susceptibility, 157, 159 Geometrical mean model, 281 Ginzburg–Landau phenomenological theory, 169 Grassmann variables, 97 Green function, 70 Green function theory, 281 Ground-state energy, 15 Ground-state magnetization, 65, 81 Ground-state sublattice magnetization, 153 Gutzwiller approximation, 43, 155 Gutzwiller overlap function, 42 Gutzwiller variational method, 40 H Hamiltonian, 105 Harmonic approximation, 90, 92, 313 Hartree–Fock approximation, 6, 31, 140, 199 Hartree–Fock cluster CPA theory, 221 Hartree–Fock CPA theory, 206, 209 Hartree–Fock energy, 32, 74 Hartree–Fock equations, 137 Hartree–Fock ground state, 118 Hartree–Fock susceptibility, 158 Hartree–Fock theory, 136 Hartree–Fock wave function, 47 Hartree–Fock–Slater exchange potential, 48 Hartree–Fock–Stoner theory, 67 Heavy effective mass, 201 Heavy fermion system, 201 Heavy rare-earth elements, 11 Heavy RE atoms, 297 Heavy transition-metal elements, 11

342 Heisenberg model, 19, 20, 22, 115, 155 Heitler–London theory, 20 Helical MSDW, 178 Helical structure, 25, 169, 179 Hilbert transform, 98 Ho, 26 Hohenberg–Kohn theorem, 49 Hohenberg–Kohn–Sham method, 51 Hubbard bands, 16 Hubbard model, 13 Hubbard–Stratonovich transformation, 68, 96, 108 Hubbard’s alloy analogy picture, 16 Hund rule, 9 Hund-rule coupling, 8, 9, 297 Hund’s first rule, 8 Hund’s second rule, 9 Hydrogen molecule, 11, 54 Hyper-cubic lattice, 97 Hypergeometric functions, 44 I Impurity action, 98, 99 Impurity Green function, 85, 96, 99 Impurity Hamiltonian, 95 Impurity susceptibility, 199 Incommensurate wave, 169, 177 Incomplete ferromagnetism, 32 Infinite dimensions, 97 Insulating state, 16 Insulator, 14 Inter-site Coulomb interaction, 13 Internal-field distribution function, 237 Intra-atomic Coulomb interaction energy, 60 Intra-atomic Coulomb interactions, 13 Intraatomic exchange interaction, 105 Invar anomalies, 268 Invar effects, 215 Irreducible representation, 5 Ising type, 226 Isothermal molecular dynamics method, 165, 323 Isotropic SG, 251 Itinerant electron magnets, 18 Itinerant ferromagnets, 118 Itinerant magnetism, 18, 30 Itinerant-electron ferromagnet, 123 Itinerant-electron SG, 271 K Kohn–Sham equation, 51 Kohn–Sham theory, 51 Kondo effects, 200 Kondo exchange coupling, 185

Index Kondo Hamiltonian, 186 Kondo lattice Hamiltonian, 185 Kondo problem, 200 Kondo temperature, 197 L Lanczos method, 325 Landau theory, 171 Landé’s g factor, 11 Laplace expansion, 313 Lattice, 151 Laves-phase Co2 Y, 292 LDA + U method, 107 Legendre transformation, 136 Ligand electron state, 195 Ligand model, 193 Light rare-earth elements, 11 Light RE atoms, 297 Light transition-metal elements, 10 Linear response, 319 Linear response function, 319 Linear response theory, 124 Linearly polarized MSDW, 177 Liouville operator, 102 Liquid Ni, 280 Liquid transition metal, 269 Local anisotropy, 274 Local ansatz variational method, 46 Local charge, 78 Local density approximation, 52, 105, 298 Local environment effects (LEE), 217, 222, 264 Local ferromagnetic orders, 271 Local magnetic moment, 11, 78, 193 Local moment system, 67, 80 Local spin density approximation, 52 Locally rotated coordinates, 164 Long-range magnetic interaction, 181 Long-wave limit, 123 Low density limit, 37 Low-energy excitations, 118 M Magnetic diffuse-scattering, 221 Magnetic entropy, 79 Magnetic force, 166 Magnetic form factor, 129 Magnetic moment, 1 Magnetic phase diagram, 155, 175 Magnetic quantum number, 5 Magnetic structure, 22 Magnetization, 66, 124, 171, 239, 268 Magnon excitations, 123 Many-body CPA, 89, 95

Index Matthiessen’s law, 200 MB-CPA, 95, 102 MD approach, 242 Mean-field theory, 67 Medium, 75 Metal, 14, 16 Metal–insulator crossover, 155 Metal–insulator transition, 16, 46 Metallic ferromagnet, 18 Metallic magnetism, 18 Metallic magnets, 18 Metalloids, 253 Mn, 18, 149, 162 Mnc Y1−c , 293 MnO, 17 MnO2 , 26 MnSi, 180 Mode–mode coupling, 92, 176, 178 Modulated structure, 169 Molecular dynamics method, 163, 256 Momentum independent self-energy, 98 Most random configuration, 281 Mott insulator, 16 Mott transition, 16 MSDW, 172, 177 Muffin-tin, 54 Multicritical point, 273 Multiple scattering theory, 305 Multiple spin density wave, 172, 177 Multiplet, 9 N Nd–Co, 297 Nd–Fe, 297 Nearly orthogonal representation, 56 Néel temperature, 154, 155 Nesting effect, 160 Nesting mechanism, 149 Nesting-type antiferromagnetism, 154 Neutral state, 12 Neutron magnetic scattering, 129 Ni, 23, 29, 35, 60, 61, 67, 81, 111, 112, 133, 160 Ni–B, 277, 280 Ni–Cr, 240 Ni–Cu, 221, 240 Ni–La, 277 Ni–Mn, 230, 240, 242 Ni–Y, 277 Ni–Zr, 277 Ni1−x Crx , 210 Ni3 Al, 135 Nic Y1−c , 293 Nickel, 40

343 NiO, 16, 17 Non-collinear ferrimagnetism, 297 Noncollinear, 265, 272 Noncollinear ferromagnetism, 266, 273, 297 Noncollinear magnetic structure, 297 Noncollinear SG, 266, 273 Noncollinear theory, 272 Nonlinear magnetic coupling, 234, 240, 270 Nonlocal susceptibility, 158 Nonunique magnetism, 277 Nuclear magneton, 3 Nuclear magneton number, 129 Numerical renormalization group method, 98 Numerical renormalization group technique, 199 O Off-diagonal disorder, 207, 208, 218 Off-diagonal Green function, 327 Off-diagonal self-energy, 98 Onsager’s reaction field, 147 Orbital degeneracy, 93, 105 Orbital magnetic moment, 2, 18 P Pair distribution function, 255 Pair energy, 225 Paramagnetic spin susceptibility, 32 Paramagnetic state, 191 Paramagnetic susceptibility, 65, 81, 171 Paramagnetism, 265, 266, 278 Path integral method, 97 Pauli principle, 8, 38, 68 Pauli spin matrices, 3 Pauli uniform susceptibility, 189 Pd, 61 Phenomenological spin-fluctuation theory, 172 PM-CPA, 102 Point group, 5 Polarized state, 12 Potential parameter, 57 Principal quantum number, 4 Projection operator, 20 Projection operator method CPA, 89, 102 Projection technique, 103 Pseudo-potential, 256 Pt, 61 Q Quantum fluctuations, 93 Quantum Monte-Carlo method, 98 Quenching of orbital magnetic moments, 18 Random phase approximation, 111, 120, 129 Rapid quenching, 253 Rare-earth metals, 18

344 Rare-earth transition metal, 296 Re-entrant SG, 272, 290 Recursion method, 167, 243, 257, 325 Relaxed DRPHS model, 256 Renormalization group theory, 201 Response function, 125 Retarded Green function, 102, 130 Rhodes–Wohlfarth ratio, 112, 135 Riemann’s ζ function, 124 Rigid band picture, 244 Rigid-band model, 204 RKKY interaction, 187, 200, 329 Rotational invariance, 164, 165 RPA, 120, 129 S Sc3 In, 135, 147 Scattering vector, 129 Screened structure constant, 55 Sd Hamiltonian, 186 SDW, 160 Second-order Born approximation, 200 Second-order transition, 215 Self-consistent phonon approximation, 147 Self-consistent renormalization theory (SCR), 145 Self-energy, 261, 307 SG, 234, 236, 244, 265, 268 SG states, 270 SG temperature, 191 Short-range atomic order, 192 Simple cubic, 151 Simple dilution picture, 240 Single-site approximation, 77 Single-site spin fluctuation theory, 75, 90, 309 Single-site theory, 74 Single-site t matrix, 76, 84 Singlet, 38, 194, 200 Singlet state, 195 Sinusoidal spin density wave, 24 Slater determinant, 8 Slater–Pauling curves, 203, 239 Slater’s two-center tight-binding Hamiltonian, 58 Sommerfeld coefficients, 29 Spherical harmonics, 5 Spin angular momentum, 2 Spin correlations, 47 Spin density waves, 160, 278 Spin disorder, 259 Spin fluctuation parameter, 145 Spin glass, 28, 234, 278 Spin glass state, 191, 200, 215, 229, 231, 287 Spin magnetic moment, 2

Index Spin susceptibility, 61 Spin wave, 115, 118, 132, 238 Spin wave energy, 117 Spin wave excitations, 121, 123, 272 Spin wave stiffness constant, 122, 132, 274 Spin–orbit coupling, 297 Spin–orbit interaction, 4, 18, 170 Spin-flip magnetic moment, 127 Spin-flip operator, 117, 119 Spin-glass temperature, 229 Spontaneous magnetization, 139 Stability condition, 174 Staggered susceptibility, 160 Static approximation, 69, 73, 163 Static energy potential, 91 Static field variables, 72 Static potential, 109 Stationary condition, 77, 86, 302 Stoner condition, 31, 35, 60, 267 Stoner continuum, 122 Stoner criterion, 160, 292 Stoner excitations, 121, 123, 132 Stoner parameter, 58 Stoner theory, 64, 135 Strong ferromagnetism, 32 Strong ferromagnets, 204 Structural disorder, 253, 259, 280, 296 Sublattice, 24, 151 Substitutional disordered alloys, 27 Sum rule, 144 Super-conducting order parameter, 153 Super-exchange interaction, 22, 155, 181, 226 Susceptibility, 11, 66 Symmetry operations, 170 T Tb, 26 TB-LMTO, 106, 257 Temperature Green function, 71, 95 Terminator, 244, 326 Ternary alloys, 251 Thermal spin fluctuations, 93 Ti, 35 Tight-binding linear muffin-tin orbital (TB-LMTO), 54, 106, 159 Tight-binding LMTO, 163 Tight-binding LMTO supercell method, 289 Time reversal symmetry, 170 Time-dependent coherent potential, 95 Time-dependent Green function, 96 Time-dependent Hamiltonian, 70 Time-dependent magnetic moment, 166 Time-dependent potential, 70 Time-ordered product, 69

Index Tm, 29 Total angular momentum, 9 Transition metal metalloid, 295 Transverse dynamical susceptibility, 127 Transverse SG order parameter, 272 Transverse spin freezing temperature, 274 Tridiagonal matrices, 314 Triple Q MSDW, 174 Triple Q multiple SDW, 27 Triplet, 38, 194 Triplet states, 195 Two-band model, 182 U U, 201 Unenhanced susceptibility, 160 Uniform susceptibility, 139 V V, 60

345 Variational approach, 104 Very weak ferromagnetism, 65 Virtual bound state, 205, 210, 240 Voronoi polyhedra, 23 W Weak ferromagnetism, 32 Weiss-field function, 98 Wigner–Seitz cells, 23 X Xα potential, 48 Z Zeeman interaction, 4 Zero-band-width model, 193 ZrZn2 , 135, 147